| L(s) = 1 | − 2·2-s − 3-s + 2·4-s − 5-s + 2·6-s − 7-s + 9-s + 2·10-s + 4·11-s − 2·12-s − 4·13-s + 2·14-s + 15-s − 4·16-s + 2·17-s − 2·18-s − 2·20-s + 21-s − 8·22-s − 7·23-s + 25-s + 8·26-s − 27-s − 2·28-s − 6·29-s − 2·30-s − 8·31-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s + 4-s − 0.447·5-s + 0.816·6-s − 0.377·7-s + 1/3·9-s + 0.632·10-s + 1.20·11-s − 0.577·12-s − 1.10·13-s + 0.534·14-s + 0.258·15-s − 16-s + 0.485·17-s − 0.471·18-s − 0.447·20-s + 0.218·21-s − 1.70·22-s − 1.45·23-s + 1/5·25-s + 1.56·26-s − 0.192·27-s − 0.377·28-s − 1.11·29-s − 0.365·30-s − 1.43·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 59 | \( 1 - T \) |
| 113 | \( 1 + T \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.35531784025453, −13.86292662999597, −13.11082212247720, −12.52710278025528, −12.16367166396146, −11.65144926239270, −11.28313252012876, −10.72279977632798, −10.15566873713051, −9.668672872282089, −9.517621128468811, −8.823375599317415, −8.317318598305679, −7.783200868260622, −7.226627710353203, −6.898114304772559, −6.423396267049531, −5.616533958778447, −5.153335960748824, −4.406843327480296, −3.767211558134807, −3.385658597216329, −2.172864560437771, −1.816414789089624, −1.070281893583117, 0, 0,
1.070281893583117, 1.816414789089624, 2.172864560437771, 3.385658597216329, 3.767211558134807, 4.406843327480296, 5.153335960748824, 5.616533958778447, 6.423396267049531, 6.898114304772559, 7.226627710353203, 7.783200868260622, 8.317318598305679, 8.823375599317415, 9.517621128468811, 9.668672872282089, 10.15566873713051, 10.72279977632798, 11.28313252012876, 11.65144926239270, 12.16367166396146, 12.52710278025528, 13.11082212247720, 13.86292662999597, 14.35531784025453
