| L(s) = 1 | + 0.866·2-s − 3-s − 1.24·4-s − 0.866·6-s + 7-s − 2.81·8-s + 9-s + 11-s + 1.24·12-s + 1.74·13-s + 0.866·14-s + 0.0589·16-s − 3.42·17-s + 0.866·18-s + 3.75·19-s − 21-s + 0.866·22-s − 5.62·23-s + 2.81·24-s + 1.51·26-s − 27-s − 1.24·28-s − 8.21·29-s + 2.54·31-s + 5.68·32-s − 33-s − 2.96·34-s + ⋯ |
| L(s) = 1 | + 0.612·2-s − 0.577·3-s − 0.624·4-s − 0.353·6-s + 0.377·7-s − 0.995·8-s + 0.333·9-s + 0.301·11-s + 0.360·12-s + 0.484·13-s + 0.231·14-s + 0.0147·16-s − 0.830·17-s + 0.204·18-s + 0.861·19-s − 0.218·21-s + 0.184·22-s − 1.17·23-s + 0.574·24-s + 0.297·26-s − 0.192·27-s − 0.236·28-s − 1.52·29-s + 0.457·31-s + 1.00·32-s − 0.174·33-s − 0.508·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 - 0.866T + 2T^{2} \) |
| 13 | \( 1 - 1.74T + 13T^{2} \) |
| 17 | \( 1 + 3.42T + 17T^{2} \) |
| 19 | \( 1 - 3.75T + 19T^{2} \) |
| 23 | \( 1 + 5.62T + 23T^{2} \) |
| 29 | \( 1 + 8.21T + 29T^{2} \) |
| 31 | \( 1 - 2.54T + 31T^{2} \) |
| 37 | \( 1 + 1.63T + 37T^{2} \) |
| 41 | \( 1 - 7.48T + 41T^{2} \) |
| 43 | \( 1 + 9.61T + 43T^{2} \) |
| 47 | \( 1 - 10.7T + 47T^{2} \) |
| 53 | \( 1 - 9.91T + 53T^{2} \) |
| 59 | \( 1 - 6.37T + 59T^{2} \) |
| 61 | \( 1 + 10.6T + 61T^{2} \) |
| 67 | \( 1 - 6.53T + 67T^{2} \) |
| 71 | \( 1 - 1.55T + 71T^{2} \) |
| 73 | \( 1 + 12.6T + 73T^{2} \) |
| 79 | \( 1 + 14.5T + 79T^{2} \) |
| 83 | \( 1 + 9.65T + 83T^{2} \) |
| 89 | \( 1 - 10.4T + 89T^{2} \) |
| 97 | \( 1 - 8.81T + 97T^{2} \) |
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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
−7.67756670129572675974435599366, −6.95838314914667821949286152286, −5.96246897833982684376940443567, −5.68770690409701384332266189961, −4.82042899784117161293461684715, −4.12805863911832467795865633014, −3.60596589898224222069121009607, −2.40679845891748102556810126978, −1.24005520809093278342952943414, 0,
1.24005520809093278342952943414, 2.40679845891748102556810126978, 3.60596589898224222069121009607, 4.12805863911832467795865633014, 4.82042899784117161293461684715, 5.68770690409701384332266189961, 5.96246897833982684376940443567, 6.95838314914667821949286152286, 7.67756670129572675974435599366
