L(s) = 1 | + 2-s + 4-s + 2·5-s + 8-s − 3·9-s + 2·10-s + 4·11-s − 2·13-s + 16-s + 2·17-s − 3·18-s − 4·19-s + 2·20-s + 4·22-s − 25-s − 2·26-s − 6·29-s + 8·31-s + 32-s + 2·34-s − 3·36-s − 2·37-s − 4·38-s + 2·40-s − 6·41-s + 4·44-s − 6·45-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s + 0.353·8-s − 9-s + 0.632·10-s + 1.20·11-s − 0.554·13-s + 1/4·16-s + 0.485·17-s − 0.707·18-s − 0.917·19-s + 0.447·20-s + 0.852·22-s − 1/5·25-s − 0.392·26-s − 1.11·29-s + 1.43·31-s + 0.176·32-s + 0.342·34-s − 1/2·36-s − 0.328·37-s − 0.648·38-s + 0.316·40-s − 0.937·41-s + 0.603·44-s − 0.894·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 254 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 254 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.024707520\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.024707520\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 127 | \( 1 + T \) |
good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−12.02136656747206538050447703783, −11.33780072958300641731250276081, −10.13832114340388631087630943607, −9.253386576859749287217727860566, −8.117249992339835855210571085729, −6.64448714332415001742882220403, −5.95974219897264538167530998765, −4.85275776393634625693918995081, −3.41339839116369540654210157483, −1.99196599088899154813992658977,
1.99196599088899154813992658977, 3.41339839116369540654210157483, 4.85275776393634625693918995081, 5.95974219897264538167530998765, 6.64448714332415001742882220403, 8.117249992339835855210571085729, 9.253386576859749287217727860566, 10.13832114340388631087630943607, 11.33780072958300641731250276081, 12.02136656747206538050447703783