| L(s) = 1 | − 2·2-s + 2·4-s − 8·7-s − 4·8-s + 3·9-s − 2·11-s + 2·13-s + 16·14-s + 8·16-s + 2·17-s − 6·18-s − 7·19-s + 4·22-s − 6·23-s − 4·26-s − 16·28-s − 9·29-s − 14·31-s − 8·32-s − 4·34-s + 6·36-s + 4·37-s + 14·38-s − 2·41-s − 2·43-s − 4·44-s + 12·46-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 3.02·7-s − 1.41·8-s + 9-s − 0.603·11-s + 0.554·13-s + 4.27·14-s + 2·16-s + 0.485·17-s − 1.41·18-s − 1.60·19-s + 0.852·22-s − 1.25·23-s − 0.784·26-s − 3.02·28-s − 1.67·29-s − 2.51·31-s − 1.41·32-s − 0.685·34-s + 36-s + 0.657·37-s + 2.27·38-s − 0.312·41-s − 0.304·43-s − 0.603·44-s + 1.76·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 19 | $C_2$ | \( 1 + 7 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 9 T + 52 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 2 T - 37 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 2 T - 39 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 4 T - 37 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 9 T + 22 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 10 T + 33 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + T - 70 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + T - 78 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 11 T + 32 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 6 T - 61 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
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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
−10.36751164086513074431839345817, −10.30600777154511825610365359512, −9.716328079682888654030215313233, −9.532024088411233542618671296969, −9.061086989972332851215588596742, −8.910473288127111086879216213944, −7.977194035414614945938543856948, −7.81680258989507878547397959837, −7.07799198823784898065570791569, −6.56685710101383599199555415969, −6.43332306229834107196970330317, −5.75222278372115589474196927489, −5.51485640114720601559545485755, −4.10228078569520378231572376488, −3.70399542926455084123582712884, −3.29013321991267835158951725753, −2.55355022563390817707266768711, −1.68095685129292630338843942599, 0, 0,
1.68095685129292630338843942599, 2.55355022563390817707266768711, 3.29013321991267835158951725753, 3.70399542926455084123582712884, 4.10228078569520378231572376488, 5.51485640114720601559545485755, 5.75222278372115589474196927489, 6.43332306229834107196970330317, 6.56685710101383599199555415969, 7.07799198823784898065570791569, 7.81680258989507878547397959837, 7.977194035414614945938543856948, 8.910473288127111086879216213944, 9.061086989972332851215588596742, 9.532024088411233542618671296969, 9.716328079682888654030215313233, 10.30600777154511825610365359512, 10.36751164086513074431839345817
