L(s) = 1 | + 4·5-s + 4·7-s − 2·9-s + 8·11-s − 4·13-s + 8·19-s + 12·23-s + 10·25-s − 8·27-s + 4·29-s − 16·31-s + 16·35-s − 4·37-s − 12·41-s − 16·43-s − 8·45-s + 8·49-s − 4·53-s + 32·55-s + 16·59-s − 4·61-s − 8·63-s − 16·65-s + 8·67-s − 12·71-s + 28·73-s + 32·77-s + ⋯ |
L(s) = 1 | + 1.78·5-s + 1.51·7-s − 2/3·9-s + 2.41·11-s − 1.10·13-s + 1.83·19-s + 2.50·23-s + 2·25-s − 1.53·27-s + 0.742·29-s − 2.87·31-s + 2.70·35-s − 0.657·37-s − 1.87·41-s − 2.43·43-s − 1.19·45-s + 8/7·49-s − 0.549·53-s + 4.31·55-s + 2.08·59-s − 0.512·61-s − 1.00·63-s − 1.98·65-s + 0.977·67-s − 1.42·71-s + 3.27·73-s + 3.64·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.560096172\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.560096172\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 2 | | \( 1 \) |
good | 3 | $C_2$$\times$$C_2^2$ | \( ( 1 - 2 T + p T^{2} )^{2}( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} ) \) |
| 5 | $C_2$$\times$$C_2^2$ | \( ( 1 - 2 T + p T^{2} )^{2}( 1 - 8 T^{2} + p^{2} T^{4} ) \) |
| 7 | $C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$$\times$$C_2^2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} ) \) |
| 13 | $D_4\times C_2$ | \( 1 + 4 T + 6 T^{2} + 4 T^{3} + 2 T^{4} + 4 p T^{5} + 6 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 8 T + 18 T^{2} + 160 T^{3} - 1246 T^{4} + 160 p T^{5} + 18 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 300 T^{3} + 1246 T^{4} - 300 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 4 T + 6 T^{2} + 204 T^{3} - 830 T^{4} + 204 p T^{5} + 6 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 37 | $D_4\times C_2$ | \( 1 + 4 T + 6 T^{2} + 4 T^{3} + 2 T^{4} + 4 p T^{5} + 6 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} + 516 T^{3} + 3694 T^{4} + 516 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 16 T + 162 T^{2} + 1384 T^{3} + 10178 T^{4} + 1384 p T^{5} + 162 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 486 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 4 T + 54 T^{2} + 708 T^{3} + 3490 T^{4} + 708 p T^{5} + 54 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 16 T + 114 T^{2} - 696 T^{3} + 4834 T^{4} - 696 p T^{5} + 114 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 4 T + 6 T^{2} + 4 T^{3} + 2 T^{4} + 4 p T^{5} + 6 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 8 T + 18 T^{2} + 736 T^{3} - 5854 T^{4} + 736 p T^{5} + 18 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} + 876 T^{3} + 10654 T^{4} + 876 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 + 16 T + 114 T^{2} + 792 T^{3} + 6370 T^{4} + 792 p T^{5} + 114 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 516 T^{3} + 1582 T^{4} - 516 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 20 T + 222 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.912136021049802459297672368006, −8.458107787022351516778618594962, −8.293628448777049616736127619548, −8.248893434841512007363450894719, −7.70271224111216286510352110144, −7.17041493581628644946594211123, −7.05900284550632611513317883738, −6.99875872112885104613691938912, −6.85095920652657877500460231579, −6.41858560037882984709383782944, −5.99150613204204100353497108807, −5.61189352190781138663255335743, −5.47768304057446132628469049816, −5.21653609939209728790409325718, −5.03023688580214743555016268227, −4.83057805339718344973478952484, −4.39331423777244873147063872420, −3.76786398912133104903730549332, −3.42440565779345375578322565239, −3.41292099597879333268552741204, −2.77988109821554120002689302271, −2.24158299734871445865888867380, −1.75815499054402504394321174642, −1.57571203772657506054788210050, −1.15571752050935234331207400243,
1.15571752050935234331207400243, 1.57571203772657506054788210050, 1.75815499054402504394321174642, 2.24158299734871445865888867380, 2.77988109821554120002689302271, 3.41292099597879333268552741204, 3.42440565779345375578322565239, 3.76786398912133104903730549332, 4.39331423777244873147063872420, 4.83057805339718344973478952484, 5.03023688580214743555016268227, 5.21653609939209728790409325718, 5.47768304057446132628469049816, 5.61189352190781138663255335743, 5.99150613204204100353497108807, 6.41858560037882984709383782944, 6.85095920652657877500460231579, 6.99875872112885104613691938912, 7.05900284550632611513317883738, 7.17041493581628644946594211123, 7.70271224111216286510352110144, 8.248893434841512007363450894719, 8.293628448777049616736127619548, 8.458107787022351516778618594962, 8.912136021049802459297672368006