| L(s) = 1 | + 2·5-s + 8·11-s − 25-s + 12·29-s − 8·31-s + 20·41-s − 2·49-s + 16·55-s + 8·59-s − 4·61-s − 24·79-s − 20·89-s − 4·101-s − 4·109-s + 26·121-s − 12·125-s + 127-s + 131-s + 137-s + 139-s + 24·145-s + 149-s + 151-s − 16·155-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 2.41·11-s − 1/5·25-s + 2.22·29-s − 1.43·31-s + 3.12·41-s − 2/7·49-s + 2.15·55-s + 1.04·59-s − 0.512·61-s − 2.70·79-s − 2.11·89-s − 0.398·101-s − 0.383·109-s + 2.36·121-s − 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.99·145-s + 0.0819·149-s + 0.0813·151-s − 1.28·155-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8294400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8294400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.197430551\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.197430551\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.129214036912626185622187726035, −8.635206160167547645352611683963, −8.387915034836467193652106471304, −7.75578452465452939134392011862, −7.44983184432701905815571031317, −6.95486517140137659624673206522, −6.56942591861431131559323606230, −6.39265695515949371628127647057, −5.98609390835085804044246900868, −5.46398715315674357324730757266, −5.35544312205131889319749782822, −4.40183455781818229239922049676, −4.19063964880481099421733463882, −4.07809651890719114961633882846, −3.30618394724812572464092514897, −2.79310678777461091084819409455, −2.39819863389881099425942861282, −1.57489083333835240207871226755, −1.40055863120825409765808011288, −0.69740321406803353008084554793,
0.69740321406803353008084554793, 1.40055863120825409765808011288, 1.57489083333835240207871226755, 2.39819863389881099425942861282, 2.79310678777461091084819409455, 3.30618394724812572464092514897, 4.07809651890719114961633882846, 4.19063964880481099421733463882, 4.40183455781818229239922049676, 5.35544312205131889319749782822, 5.46398715315674357324730757266, 5.98609390835085804044246900868, 6.39265695515949371628127647057, 6.56942591861431131559323606230, 6.95486517140137659624673206522, 7.44983184432701905815571031317, 7.75578452465452939134392011862, 8.387915034836467193652106471304, 8.635206160167547645352611683963, 9.129214036912626185622187726035
