| L(s) = 1 | + 1.42·3-s − 3.52·5-s − 1.45·7-s − 0.978·9-s + 2.46·11-s − 0.945·13-s − 5.01·15-s + 17-s − 3.98·19-s − 2.06·21-s + 8.79·23-s + 7.43·25-s − 5.65·27-s + 3.21·29-s + 8.26·31-s + 3.49·33-s + 5.11·35-s + 1.00·37-s − 1.34·39-s − 0.565·41-s + 3.42·43-s + 3.45·45-s − 11.2·47-s − 4.89·49-s + 1.42·51-s − 4.73·53-s − 8.67·55-s + ⋯ |
| L(s) = 1 | + 0.820·3-s − 1.57·5-s − 0.548·7-s − 0.326·9-s + 0.741·11-s − 0.262·13-s − 1.29·15-s + 0.242·17-s − 0.914·19-s − 0.450·21-s + 1.83·23-s + 1.48·25-s − 1.08·27-s + 0.596·29-s + 1.48·31-s + 0.608·33-s + 0.865·35-s + 0.164·37-s − 0.215·39-s − 0.0882·41-s + 0.521·43-s + 0.514·45-s − 1.64·47-s − 0.699·49-s + 0.199·51-s − 0.650·53-s − 1.16·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 - 1.42T + 3T^{2} \) |
| 5 | \( 1 + 3.52T + 5T^{2} \) |
| 7 | \( 1 + 1.45T + 7T^{2} \) |
| 11 | \( 1 - 2.46T + 11T^{2} \) |
| 13 | \( 1 + 0.945T + 13T^{2} \) |
| 19 | \( 1 + 3.98T + 19T^{2} \) |
| 23 | \( 1 - 8.79T + 23T^{2} \) |
| 29 | \( 1 - 3.21T + 29T^{2} \) |
| 31 | \( 1 - 8.26T + 31T^{2} \) |
| 37 | \( 1 - 1.00T + 37T^{2} \) |
| 41 | \( 1 + 0.565T + 41T^{2} \) |
| 43 | \( 1 - 3.42T + 43T^{2} \) |
| 47 | \( 1 + 11.2T + 47T^{2} \) |
| 53 | \( 1 + 4.73T + 53T^{2} \) |
| 61 | \( 1 - 9.48T + 61T^{2} \) |
| 67 | \( 1 - 2.29T + 67T^{2} \) |
| 71 | \( 1 - 10.2T + 71T^{2} \) |
| 73 | \( 1 - 5.35T + 73T^{2} \) |
| 79 | \( 1 + 7.48T + 79T^{2} \) |
| 83 | \( 1 + 8.32T + 83T^{2} \) |
| 89 | \( 1 + 10.7T + 89T^{2} \) |
| 97 | \( 1 - 4.97T + 97T^{2} \) |
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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
−7.67276654200610332007130226185, −6.75480779907227189438424215632, −6.49230900779009811646335467423, −5.20322946793698704878580713584, −4.45308958805470940172264746120, −3.77960172733367725099484283861, −3.13624122913390191627850394382, −2.59601493980173442996760924228, −1.14006203272547792063975895127, 0,
1.14006203272547792063975895127, 2.59601493980173442996760924228, 3.13624122913390191627850394382, 3.77960172733367725099484283861, 4.45308958805470940172264746120, 5.20322946793698704878580713584, 6.49230900779009811646335467423, 6.75480779907227189438424215632, 7.67276654200610332007130226185
