| L(s) = 1 | − 3·2-s − 3·3-s + 4·4-s − 2·5-s + 9·6-s − 4·7-s − 3·8-s + 2·9-s + 6·10-s − 7·11-s − 12·12-s − 10·13-s + 12·14-s + 6·15-s + 3·16-s − 12·17-s − 6·18-s − 8·20-s + 12·21-s + 21·22-s − 23-s + 9·24-s + 3·25-s + 30·26-s + 6·27-s − 16·28-s − 3·29-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 1.73·3-s + 2·4-s − 0.894·5-s + 3.67·6-s − 1.51·7-s − 1.06·8-s + 2/3·9-s + 1.89·10-s − 2.11·11-s − 3.46·12-s − 2.77·13-s + 3.20·14-s + 1.54·15-s + 3/4·16-s − 2.91·17-s − 1.41·18-s − 1.78·20-s + 2.61·21-s + 4.47·22-s − 0.208·23-s + 1.83·24-s + 3/5·25-s + 5.88·26-s + 1.15·27-s − 3.02·28-s − 0.557·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3258025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3258025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 19 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 3 | $D_{4}$ | \( 1 + p T + 7 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 4 T + 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 7 T + 3 p T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 + T + 45 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 3 T + 49 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 15 T + 107 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 3 T + 65 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 43 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - T + 75 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 16 T + 153 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 9 T + 125 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 11 T + 117 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 + 4 T + 93 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 12 T + 173 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 146 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 133 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + T + 163 T^{2} + p T^{3} + p^{2} T^{4} \) |
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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
−8.791639999399905009031876044644, −8.370958914666709128461865326637, −7.81080219051163735509560428592, −7.62811818360444885928886493354, −7.07648249481565671682471478050, −6.94453987324061569269805639883, −6.51909181562675383555232123008, −5.95789702411673622571777734279, −5.35145293167089716622401256017, −5.25075151174998249865816272191, −4.67976170456253772045163608928, −4.30492501472635532056353905389, −3.35158689502235047724143842977, −2.83467863339657275170986967195, −2.43038997827029182387329314482, −1.76903145686835238146365551917, 0, 0, 0, 0,
1.76903145686835238146365551917, 2.43038997827029182387329314482, 2.83467863339657275170986967195, 3.35158689502235047724143842977, 4.30492501472635532056353905389, 4.67976170456253772045163608928, 5.25075151174998249865816272191, 5.35145293167089716622401256017, 5.95789702411673622571777734279, 6.51909181562675383555232123008, 6.94453987324061569269805639883, 7.07648249481565671682471478050, 7.62811818360444885928886493354, 7.81080219051163735509560428592, 8.370958914666709128461865326637, 8.791639999399905009031876044644
