| L(s) = 1 | + 2·2-s + 3·4-s − 2·5-s + 4·8-s − 4·10-s + 4·11-s + 5·16-s − 4·17-s − 10·19-s − 6·20-s + 8·22-s − 2·23-s − 25-s − 4·29-s − 12·31-s + 6·32-s − 8·34-s + 4·37-s − 20·38-s − 8·40-s + 4·43-s + 12·44-s − 4·46-s − 2·50-s − 12·53-s − 8·55-s − 8·58-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s − 0.894·5-s + 1.41·8-s − 1.26·10-s + 1.20·11-s + 5/4·16-s − 0.970·17-s − 2.29·19-s − 1.34·20-s + 1.70·22-s − 0.417·23-s − 1/5·25-s − 0.742·29-s − 2.15·31-s + 1.06·32-s − 1.37·34-s + 0.657·37-s − 3.24·38-s − 1.26·40-s + 0.609·43-s + 1.80·44-s − 0.589·46-s − 0.282·50-s − 1.64·53-s − 1.07·55-s − 1.05·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63011844 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63011844 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 + 2 T + p T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 + 10 T + 3 p T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 4 T - 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 66 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 12 T + 118 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + 18 T + 197 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 16 T + 174 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 10 T + 143 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 4 T + 126 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 6 T + 143 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 - 24 T + 298 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4 T + 174 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−7.46390098915774958572143896262, −7.45880147801527464794137817086, −6.65474745270596768077796784290, −6.58858605544507636104263799004, −6.26752004680470982396333670465, −6.12170668273337810799685535146, −5.37643048744343971732909564859, −5.31902490421400534489453371966, −4.67320271014997143750005812456, −4.30015706201520863562864498460, −4.02832890493040470119954172101, −4.02265510168622853638485132618, −3.46515772261459680127949075129, −3.12476853424579195472593643044, −2.40286560417229822479723117573, −2.20167217404149767743655406362, −1.67109165971784665491562498218, −1.29900280346099564505913401860, 0, 0,
1.29900280346099564505913401860, 1.67109165971784665491562498218, 2.20167217404149767743655406362, 2.40286560417229822479723117573, 3.12476853424579195472593643044, 3.46515772261459680127949075129, 4.02265510168622853638485132618, 4.02832890493040470119954172101, 4.30015706201520863562864498460, 4.67320271014997143750005812456, 5.31902490421400534489453371966, 5.37643048744343971732909564859, 6.12170668273337810799685535146, 6.26752004680470982396333670465, 6.58858605544507636104263799004, 6.65474745270596768077796784290, 7.45880147801527464794137817086, 7.46390098915774958572143896262
