L(s) = 1 | + 2-s + 4-s + 8-s − 2·11-s + 4·13-s + 16-s − 2·17-s − 2·22-s − 6·23-s + 4·26-s − 2·29-s + 4·31-s + 32-s − 2·34-s − 8·37-s + 6·41-s + 8·43-s − 2·44-s − 6·46-s − 6·47-s − 7·49-s + 4·52-s + 6·53-s − 2·58-s + 4·59-s − 6·61-s + 4·62-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s − 0.603·11-s + 1.10·13-s + 1/4·16-s − 0.485·17-s − 0.426·22-s − 1.25·23-s + 0.784·26-s − 0.371·29-s + 0.718·31-s + 0.176·32-s − 0.342·34-s − 1.31·37-s + 0.937·41-s + 1.21·43-s − 0.301·44-s − 0.884·46-s − 0.875·47-s − 49-s + 0.554·52-s + 0.824·53-s − 0.262·58-s + 0.520·59-s − 0.768·61-s + 0.508·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−13.43025304533598, −13.17524990588950, −12.46467830580362, −12.31530931337385, −11.45709017324978, −11.34574380712902, −10.62282863504332, −10.39007942660687, −9.761538435042985, −9.168962601409890, −8.619664095024031, −8.136394721164649, −7.688568717195640, −7.141684208615930, −6.477719779770887, −6.090632048148794, −5.708838027731677, −4.991549669973795, −4.602361565075214, −3.838888753535992, −3.621846953800584, −2.840671892331618, −2.255183746211516, −1.709074518741348, −0.9162250964773129, 0,
0.9162250964773129, 1.709074518741348, 2.255183746211516, 2.840671892331618, 3.621846953800584, 3.838888753535992, 4.602361565075214, 4.991549669973795, 5.708838027731677, 6.090632048148794, 6.477719779770887, 7.141684208615930, 7.688568717195640, 8.136394721164649, 8.619664095024031, 9.168962601409890, 9.761538435042985, 10.39007942660687, 10.62282863504332, 11.34574380712902, 11.45709017324978, 12.31530931337385, 12.46467830580362, 13.17524990588950, 13.43025304533598