| L(s) = 1 | + 3-s − 2·9-s − 2·11-s + 3·13-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 − 12·53-s − 4·59-s + 6·61-s − 12·67-s − 69-s + 13·71-s − 3·73-s − 5·75-s + 4·79-s + 81-s − 16·83-s + 87-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 2/3·9-s − 0.603·11-s + 0.832·13-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 − 1.64·53-s − 0.520·59-s + 0.768·61-s − 1.46·67-s − 0.120·69-s + 1.54·71-s − 0.351·73-s − 0.577·75-s + 0.450·79-s + 1/9·81-s − 1.75·83-s + 0.107·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4508 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4508 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 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
−8.178714934143292754633631975841, −7.42895605582740666955064258649, −6.43909559746313255085418462566, −5.82695727891133290421111337861, −5.04457493600001367164004189030, −4.05533155888510444529753073711, −3.27576396767122072553439631725, −2.54509720536420882221940616487, −1.52971166928607160209544360363, 0,
1.52971166928607160209544360363, 2.54509720536420882221940616487, 3.27576396767122072553439631725, 4.05533155888510444529753073711, 5.04457493600001367164004189030, 5.82695727891133290421111337861, 6.43909559746313255085418462566, 7.42895605582740666955064258649, 8.178714934143292754633631975841
