L(s) = 1 | + 2-s + 4-s + 5-s + 7-s + 8-s + 10-s + 11-s − 4·13-s + 14-s + 16-s − 2·17-s − 6·19-s + 20-s + 22-s − 6·23-s + 25-s − 4·26-s + 28-s + 2·29-s + 32-s − 2·34-s + 35-s − 10·37-s − 6·38-s + 40-s − 6·41-s − 2·43-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s + 0.353·8-s + 0.316·10-s + 0.301·11-s − 1.10·13-s + 0.267·14-s + 1/4·16-s − 0.485·17-s − 1.37·19-s + 0.223·20-s + 0.213·22-s − 1.25·23-s + 1/5·25-s − 0.784·26-s + 0.188·28-s + 0.371·29-s + 0.176·32-s − 0.342·34-s + 0.169·35-s − 1.64·37-s − 0.973·38-s + 0.158·40-s − 0.937·41-s − 0.304·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
good | 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 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
−7.47247584740724343266406040873, −6.70885569124247030567546825205, −6.25450293223026577224050347863, −5.37322770219301841811290197588, −4.73741107868240282529078108754, −4.14143675145417043094613613697, −3.19774371523316717403326899408, −2.19845413269873402875514143691, −1.72634924628833386403940971413, 0,
1.72634924628833386403940971413, 2.19845413269873402875514143691, 3.19774371523316717403326899408, 4.14143675145417043094613613697, 4.73741107868240282529078108754, 5.37322770219301841811290197588, 6.25450293223026577224050347863, 6.70885569124247030567546825205, 7.47247584740724343266406040873