L(s) = 1 | + 2-s + 4-s + 5-s + 2·7-s + 8-s + 10-s − 6·11-s + 4·13-s + 2·14-s + 16-s + 6·17-s + 20-s − 6·22-s + 25-s + 4·26-s + 2·28-s − 8·31-s + 32-s + 6·34-s + 2·35-s − 8·37-s + 40-s − 12·41-s + 2·43-s − 6·44-s − 3·49-s + 50-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.755·7-s + 0.353·8-s + 0.316·10-s − 1.80·11-s + 1.10·13-s + 0.534·14-s + 1/4·16-s + 1.45·17-s + 0.223·20-s − 1.27·22-s + 1/5·25-s + 0.784·26-s + 0.377·28-s − 1.43·31-s + 0.176·32-s + 1.02·34-s + 0.338·35-s − 1.31·37-s + 0.158·40-s − 1.87·41-s + 0.304·43-s − 0.904·44-s − 3/7·49-s + 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 19 | \( 1 \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−15.32494984100819, −14.66707403122035, −14.11367709057587, −13.84876908990929, −13.11976243783588, −12.84323879338422, −12.25796460626554, −11.60132963400156, −11.00633183793675, −10.60643071829194, −10.16355875757420, −9.515413945248164, −8.611082474302622, −8.198033225194587, −7.675068196004022, −7.111357243245250, −6.342621752714216, −5.615040380869830, −5.285933679001068, −4.923884544954563, −3.922561444170268, −3.308252264085421, −2.766545758770210, −1.788996173129138, −1.392276886325065, 0,
1.392276886325065, 1.788996173129138, 2.766545758770210, 3.308252264085421, 3.922561444170268, 4.923884544954563, 5.285933679001068, 5.615040380869830, 6.342621752714216, 7.111357243245250, 7.675068196004022, 8.198033225194587, 8.611082474302622, 9.515413945248164, 10.16355875757420, 10.60643071829194, 11.00633183793675, 11.60132963400156, 12.25796460626554, 12.84323879338422, 13.11976243783588, 13.84876908990929, 14.11367709057587, 14.66707403122035, 15.32494984100819