| L(s) = 1 | − 3-s + 9-s − 6·11-s − 2·13-s + 4·19-s − 6·23-s − 5·25-s − 27-s + 6·29-s − 8·31-s + 6·33-s + 2·37-s + 2·39-s − 12·41-s − 4·43-s − 12·47-s − 6·53-s − 4·57-s + 10·61-s + 8·67-s + 6·69-s + 6·71-s + 10·73-s + 5·75-s − 4·79-s + 81-s + 12·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 1.80·11-s − 0.554·13-s + 0.917·19-s − 1.25·23-s − 25-s − 0.192·27-s + 1.11·29-s − 1.43·31-s + 1.04·33-s + 0.328·37-s + 0.320·39-s − 1.87·41-s − 0.609·43-s − 1.75·47-s − 0.824·53-s − 0.529·57-s + 1.28·61-s + 0.977·67-s + 0.722·69-s + 0.712·71-s + 1.17·73-s + 0.577·75-s − 0.450·79-s + 1/9·81-s + 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 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
−10.11620249108594953836248897914, −9.790967618641863520242547413629, −8.245440388686575346900612009742, −7.67615829141164438776279339720, −6.60712751651675961597255577118, −5.45823535038952231993330195200, −4.92172543859600215194027340087, −3.44705194784191488479832061991, −2.08973766081077835946990875259, 0,
2.08973766081077835946990875259, 3.44705194784191488479832061991, 4.92172543859600215194027340087, 5.45823535038952231993330195200, 6.60712751651675961597255577118, 7.67615829141164438776279339720, 8.245440388686575346900612009742, 9.790967618641863520242547413629, 10.11620249108594953836248897914
