| L(s) = 1 | − 1.52·3-s + 5-s − 0.329·7-s − 0.670·9-s + 0.879·11-s − 1.35·13-s − 1.52·15-s + 5.60·17-s − 19-s + 0.502·21-s − 8.77·23-s + 25-s + 5.60·27-s − 2.54·29-s + 0.394·31-s − 1.34·33-s − 0.329·35-s + 9.34·37-s + 2.06·39-s − 3.05·41-s − 4.22·43-s − 0.670·45-s − 1.51·47-s − 6.89·49-s − 8.54·51-s + 0.669·53-s + 0.879·55-s + ⋯ |
| L(s) = 1 | − 0.881·3-s + 0.447·5-s − 0.124·7-s − 0.223·9-s + 0.265·11-s − 0.375·13-s − 0.394·15-s + 1.35·17-s − 0.229·19-s + 0.109·21-s − 1.82·23-s + 0.200·25-s + 1.07·27-s − 0.473·29-s + 0.0707·31-s − 0.233·33-s − 0.0556·35-s + 1.53·37-s + 0.330·39-s − 0.476·41-s − 0.643·43-s − 0.100·45-s − 0.220·47-s − 0.984·49-s − 1.19·51-s + 0.0919·53-s + 0.118·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 1.52T + 3T^{2} \) |
| 7 | \( 1 + 0.329T + 7T^{2} \) |
| 11 | \( 1 - 0.879T + 11T^{2} \) |
| 13 | \( 1 + 1.35T + 13T^{2} \) |
| 17 | \( 1 - 5.60T + 17T^{2} \) |
| 23 | \( 1 + 8.77T + 23T^{2} \) |
| 29 | \( 1 + 2.54T + 29T^{2} \) |
| 31 | \( 1 - 0.394T + 31T^{2} \) |
| 37 | \( 1 - 9.34T + 37T^{2} \) |
| 41 | \( 1 + 3.05T + 41T^{2} \) |
| 43 | \( 1 + 4.22T + 43T^{2} \) |
| 47 | \( 1 + 1.51T + 47T^{2} \) |
| 53 | \( 1 - 0.669T + 53T^{2} \) |
| 59 | \( 1 + 0.155T + 59T^{2} \) |
| 61 | \( 1 - 9.95T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 + 3.29T + 71T^{2} \) |
| 73 | \( 1 - 0.445T + 73T^{2} \) |
| 79 | \( 1 + 14.4T + 79T^{2} \) |
| 83 | \( 1 - 0.220T + 83T^{2} \) |
| 89 | \( 1 - 4.04T + 89T^{2} \) |
| 97 | \( 1 - 5.19T + 97T^{2} \) |
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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
−8.250631447385302360866630123641, −7.61933298638585188908047955931, −6.58571277657294753186723663404, −5.98636658114882588747438976420, −5.46731447417251298439566627328, −4.58994491386041966716205870162, −3.58693969579463836490442370877, −2.54109357820725989505537620387, −1.36403492637795508376909165950, 0,
1.36403492637795508376909165950, 2.54109357820725989505537620387, 3.58693969579463836490442370877, 4.58994491386041966716205870162, 5.46731447417251298439566627328, 5.98636658114882588747438976420, 6.58571277657294753186723663404, 7.61933298638585188908047955931, 8.250631447385302360866630123641
