| L(s) = 1 | − 2·2-s + 3·4-s − 2·5-s + 7-s − 4·8-s + 4·10-s − 3·11-s − 2·13-s − 2·14-s + 5·16-s − 9·17-s + 19-s − 6·20-s + 6·22-s − 3·23-s + 3·25-s + 4·26-s + 3·28-s − 3·29-s − 2·31-s − 6·32-s + 18·34-s − 2·35-s − 8·37-s − 2·38-s + 8·40-s − 6·41-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 0.894·5-s + 0.377·7-s − 1.41·8-s + 1.26·10-s − 0.904·11-s − 0.554·13-s − 0.534·14-s + 5/4·16-s − 2.18·17-s + 0.229·19-s − 1.34·20-s + 1.27·22-s − 0.625·23-s + 3/5·25-s + 0.784·26-s + 0.566·28-s − 0.557·29-s − 0.359·31-s − 1.06·32-s + 3.08·34-s − 0.338·35-s − 1.31·37-s − 0.324·38-s + 1.26·40-s − 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 - T + 6 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 3 T + 16 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 9 T + 46 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - T + 30 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 3 T + 40 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 3 T + 52 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 2 T + 30 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 + 17 T + 150 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 9 T + 106 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 3 T + 112 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - T + 48 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 11 T + 102 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 - 9 T + 178 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 15 T + 160 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 11 T + 216 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
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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
−9.964897810506193403771073845610, −9.722222074412542969124389187578, −8.918752254771899341705025958081, −8.838423315300725977899037736193, −8.373148729430320810238110531522, −7.972922304136515581376995405578, −7.63206297179919225413092809520, −7.15873190377694161348837291428, −6.60774742832666562931735721353, −6.57940475916617256346449120066, −5.63223326979706491609826453510, −5.08857654058536169427778392372, −4.69832201779512502154055045339, −4.07993618392966592820233166127, −3.27885453626186927846118037132, −2.91078523707722402989314214538, −1.90865886111294544803438743625, −1.76100194264193643919345998440, 0, 0,
1.76100194264193643919345998440, 1.90865886111294544803438743625, 2.91078523707722402989314214538, 3.27885453626186927846118037132, 4.07993618392966592820233166127, 4.69832201779512502154055045339, 5.08857654058536169427778392372, 5.63223326979706491609826453510, 6.57940475916617256346449120066, 6.60774742832666562931735721353, 7.15873190377694161348837291428, 7.63206297179919225413092809520, 7.972922304136515581376995405578, 8.373148729430320810238110531522, 8.838423315300725977899037736193, 8.918752254771899341705025958081, 9.722222074412542969124389187578, 9.964897810506193403771073845610
