| L(s) = 1 | − 2·3-s + 4-s + 9-s − 6·11-s − 2·12-s + 8·17-s + 6·19-s + 2·27-s + 4·29-s + 12·33-s + 36-s + 24·37-s + 12·43-s − 6·44-s − 5·49-s − 16·51-s + 8·53-s − 12·57-s − 12·59-s + 8·61-s − 64-s − 12·67-s + 8·68-s − 24·71-s + 6·76-s + 40·79-s − 4·81-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/2·4-s + 1/3·9-s − 1.80·11-s − 0.577·12-s + 1.94·17-s + 1.37·19-s + 0.384·27-s + 0.742·29-s + 2.08·33-s + 1/6·36-s + 3.94·37-s + 1.82·43-s − 0.904·44-s − 5/7·49-s − 2.24·51-s + 1.09·53-s − 1.58·57-s − 1.56·59-s + 1.02·61-s − 1/8·64-s − 1.46·67-s + 0.970·68-s − 2.84·71-s + 0.688·76-s + 4.50·79-s − 4/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.185915771\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.185915771\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 3 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_2^2$ | \( 1 + 23 T^{2} + p^{2} T^{4} \) |
| good | 7 | $C_2^3$ | \( 1 + 5 T^{2} - 24 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 6 T + 3 p T^{2} + 126 T^{3} + 452 T^{4} + 126 p T^{5} + 3 p^{3} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 - 4 T - T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 6 T + 37 T^{2} - 150 T^{3} + 492 T^{4} - 150 p T^{5} + 37 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2^3$ | \( 1 - 34 T^{2} + 627 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 4 T - 34 T^{2} + 32 T^{3} + 1195 T^{4} + 32 p T^{5} - 34 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 1254 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 24 T + 313 T^{2} - 2904 T^{3} + 20376 T^{4} - 2904 p T^{5} + 313 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^3$ | \( 1 + 66 T^{2} + 2675 T^{4} + 66 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 - 6 T - 7 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 146 T^{2} + 9315 T^{4} - 146 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 4 T + 107 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 12 T + 114 T^{2} + 792 T^{3} + 3707 T^{4} + 792 p T^{5} + 114 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 8 T - 62 T^{2} - 32 T^{3} + 8251 T^{4} - 32 p T^{5} - 62 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 12 T + 190 T^{2} + 1704 T^{3} + 18891 T^{4} + 1704 p T^{5} + 190 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 24 T + 346 T^{2} + 3696 T^{3} + 32307 T^{4} + 3696 p T^{5} + 346 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 - 98 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 20 T + 210 T^{2} - 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 236 T^{2} + 25974 T^{4} - 236 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 42 T + 909 T^{2} + 13482 T^{3} + 147452 T^{4} + 13482 p T^{5} + 909 p^{2} T^{6} + 42 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 36 T + 730 T^{2} + 10728 T^{3} + 121299 T^{4} + 10728 p T^{5} + 730 p^{2} T^{6} + 36 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.41312899975011534619386835607, −6.19989001941826674503214848193, −6.06753888335347049250698798736, −5.97708714276224548819047711842, −5.51143858477030856745526484842, −5.47288094243062781856749572738, −5.36852845469605457121441261892, −5.17611494431762045748514054102, −5.11725511031364629754326143424, −4.47622025072086611856675238705, −4.47351803992576777438824091026, −4.11710770944978326839213544625, −3.95308003957921175125626079283, −3.94223996729133162098315367672, −3.20621140500830928213433889845, −2.89163106274685130148448249652, −2.83673687110626820332320074019, −2.78220549158276355983459502164, −2.69862954522898376000592264296, −2.16982683995393301914996609474, −1.57295866413592374757341796826, −1.40157107495633824316268577691, −1.11332001141309890380702931459, −0.71488147486652170105763602358, −0.35659063836957216344422450850,
0.35659063836957216344422450850, 0.71488147486652170105763602358, 1.11332001141309890380702931459, 1.40157107495633824316268577691, 1.57295866413592374757341796826, 2.16982683995393301914996609474, 2.69862954522898376000592264296, 2.78220549158276355983459502164, 2.83673687110626820332320074019, 2.89163106274685130148448249652, 3.20621140500830928213433889845, 3.94223996729133162098315367672, 3.95308003957921175125626079283, 4.11710770944978326839213544625, 4.47351803992576777438824091026, 4.47622025072086611856675238705, 5.11725511031364629754326143424, 5.17611494431762045748514054102, 5.36852845469605457121441261892, 5.47288094243062781856749572738, 5.51143858477030856745526484842, 5.97708714276224548819047711842, 6.06753888335347049250698798736, 6.19989001941826674503214848193, 6.41312899975011534619386835607
