| L(s) = 1 | − 2-s − 4-s − 2·5-s + 3·8-s + 2·10-s − 11-s + 2·13-s − 16-s − 2·17-s + 2·20-s + 22-s − 8·23-s − 25-s − 2·26-s + 6·29-s + 8·31-s − 5·32-s + 2·34-s + 6·37-s − 6·40-s − 2·41-s + 44-s + 8·46-s + 8·47-s + 50-s − 2·52-s − 6·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.894·5-s + 1.06·8-s + 0.632·10-s − 0.301·11-s + 0.554·13-s − 1/4·16-s − 0.485·17-s + 0.447·20-s + 0.213·22-s − 1.66·23-s − 1/5·25-s − 0.392·26-s + 1.11·29-s + 1.43·31-s − 0.883·32-s + 0.342·34-s + 0.986·37-s − 0.948·40-s − 0.312·41-s + 0.150·44-s + 1.17·46-s + 1.16·47-s + 0.141·50-s − 0.277·52-s − 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4851 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4851 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.092649848612792766954418771551, −7.55113610427119386648970831546, −6.55118736577760269645886929040, −5.83278296173294764308357143188, −4.65970646396455232760718697930, −4.28499925734034658438358048834, −3.43632280535116392331032608080, −2.26752106768048411048763527476, −1.03326922256959689494698211571, 0,
1.03326922256959689494698211571, 2.26752106768048411048763527476, 3.43632280535116392331032608080, 4.28499925734034658438358048834, 4.65970646396455232760718697930, 5.83278296173294764308357143188, 6.55118736577760269645886929040, 7.55113610427119386648970831546, 8.092649848612792766954418771551
