| L(s) = 1 | − 2-s − 4-s + 2·5-s − 7-s + 3·8-s − 2·10-s − 6·11-s + 13-s + 14-s − 16-s + 5·19-s − 2·20-s + 6·22-s + 2·23-s − 25-s − 26-s + 28-s + 6·29-s − 7·31-s − 5·32-s − 2·35-s − 7·37-s − 5·38-s + 6·40-s + 6·41-s + 7·43-s + 6·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.894·5-s − 0.377·7-s + 1.06·8-s − 0.632·10-s − 1.80·11-s + 0.277·13-s + 0.267·14-s − 1/4·16-s + 1.14·19-s − 0.447·20-s + 1.27·22-s + 0.417·23-s − 1/5·25-s − 0.196·26-s + 0.188·28-s + 1.11·29-s − 1.25·31-s − 0.883·32-s − 0.338·35-s − 1.15·37-s − 0.811·38-s + 0.948·40-s + 0.937·41-s + 1.06·43-s + 0.904·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−8.610669458588447357989850456492, −7.78521020073562122444728676100, −7.27333043666983284665548920512, −6.09334064493131882053312075649, −5.34125161449352351656292089800, −4.79216426991236760204650994159, −3.46988814885606600552792853543, −2.51976613306045938193861212532, −1.36741406968411664209091370409, 0,
1.36741406968411664209091370409, 2.51976613306045938193861212532, 3.46988814885606600552792853543, 4.79216426991236760204650994159, 5.34125161449352351656292089800, 6.09334064493131882053312075649, 7.27333043666983284665548920512, 7.78521020073562122444728676100, 8.610669458588447357989850456492
