| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 3·7-s + 8-s + 9-s − 5·11-s − 12-s + 3·13-s + 3·14-s + 16-s + 18-s + 7·19-s − 3·21-s − 5·22-s + 4·23-s − 24-s + 3·26-s − 27-s + 3·28-s + 29-s − 6·31-s + 32-s + 5·33-s + 36-s + 6·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 1.13·7-s + 0.353·8-s + 1/3·9-s − 1.50·11-s − 0.288·12-s + 0.832·13-s + 0.801·14-s + 1/4·16-s + 0.235·18-s + 1.60·19-s − 0.654·21-s − 1.06·22-s + 0.834·23-s − 0.204·24-s + 0.588·26-s − 0.192·27-s + 0.566·28-s + 0.185·29-s − 1.07·31-s + 0.176·32-s + 0.870·33-s + 1/6·36-s + 0.986·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.141294070\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.141294070\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 43 | \( 1 + T \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−7.68659461444916208581345263246, −7.48923743235337785228826286207, −6.43872025122153156691519928834, −5.64882040950947449976348434646, −5.12335176682115789733342155595, −4.72923501142861504864485699105, −3.65754487627648625877611438441, −2.86960787427151184933710343589, −1.84389550511558348133994556922, −0.885967562051183954916667892914,
0.885967562051183954916667892914, 1.84389550511558348133994556922, 2.86960787427151184933710343589, 3.65754487627648625877611438441, 4.72923501142861504864485699105, 5.12335176682115789733342155595, 5.64882040950947449976348434646, 6.43872025122153156691519928834, 7.48923743235337785228826286207, 7.68659461444916208581345263246
