L(s) = 1 | − 3-s + 7-s − 2·9-s − 2·11-s − 3·13-s − 21-s − 23-s − 5·25-s + 5·27-s + 29-s − 5·31-s + 2·33-s − 8·37-s + 3·39-s − 7·41-s − 4·43-s + 3·47-s + 49-s − 12·53-s + 4·59-s − 6·61-s − 2·63-s − 12·67-s + 69-s + 13·71-s + 3·73-s + 5·75-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.377·7-s − 2/3·9-s − 0.603·11-s − 0.832·13-s − 0.218·21-s − 0.208·23-s − 25-s + 0.962·27-s + 0.185·29-s − 0.898·31-s + 0.348·33-s − 1.31·37-s + 0.480·39-s − 1.09·41-s − 0.609·43-s + 0.437·47-s + 1/7·49-s − 1.64·53-s + 0.520·59-s − 0.768·61-s − 0.251·63-s − 1.46·67-s + 0.120·69-s + 1.54·71-s + 0.351·73-s + 0.577·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 13 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 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
−10.30121697796105237135221389128, −9.314485556914391402538066342551, −8.306749636481555894224848604092, −7.52782061682712386881135546227, −6.45806797842396431162497925094, −5.44771453967952110851785410965, −4.82854198389058525598511373825, −3.37441439550896189970755384049, −2.04286531868059728374706246622, 0,
2.04286531868059728374706246622, 3.37441439550896189970755384049, 4.82854198389058525598511373825, 5.44771453967952110851785410965, 6.45806797842396431162497925094, 7.52782061682712386881135546227, 8.306749636481555894224848604092, 9.314485556914391402538066342551, 10.30121697796105237135221389128