| L(s) = 1 | + 2.49·3-s − 2.70·5-s + 3.48·7-s + 3.20·9-s − 4.95·11-s − 1.54·13-s − 6.74·15-s + 1.95·17-s − 4.64·19-s + 8.67·21-s + 2.32·25-s + 0.516·27-s + 0.0794·29-s − 6.97·31-s − 12.3·33-s − 9.42·35-s − 11.5·37-s − 3.84·39-s + 8.96·41-s − 9.79·43-s − 8.67·45-s + 7.58·47-s + 5.12·49-s + 4.87·51-s − 9.11·53-s + 13.4·55-s − 11.5·57-s + ⋯ |
| L(s) = 1 | + 1.43·3-s − 1.21·5-s + 1.31·7-s + 1.06·9-s − 1.49·11-s − 0.428·13-s − 1.74·15-s + 0.474·17-s − 1.06·19-s + 1.89·21-s + 0.464·25-s + 0.0993·27-s + 0.0147·29-s − 1.25·31-s − 2.14·33-s − 1.59·35-s − 1.89·37-s − 0.615·39-s + 1.39·41-s − 1.49·43-s − 1.29·45-s + 1.10·47-s + 0.732·49-s + 0.682·51-s − 1.25·53-s + 1.80·55-s − 1.53·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 - 2.49T + 3T^{2} \) |
| 5 | \( 1 + 2.70T + 5T^{2} \) |
| 7 | \( 1 - 3.48T + 7T^{2} \) |
| 11 | \( 1 + 4.95T + 11T^{2} \) |
| 13 | \( 1 + 1.54T + 13T^{2} \) |
| 17 | \( 1 - 1.95T + 17T^{2} \) |
| 19 | \( 1 + 4.64T + 19T^{2} \) |
| 29 | \( 1 - 0.0794T + 29T^{2} \) |
| 31 | \( 1 + 6.97T + 31T^{2} \) |
| 37 | \( 1 + 11.5T + 37T^{2} \) |
| 41 | \( 1 - 8.96T + 41T^{2} \) |
| 43 | \( 1 + 9.79T + 43T^{2} \) |
| 47 | \( 1 - 7.58T + 47T^{2} \) |
| 53 | \( 1 + 9.11T + 53T^{2} \) |
| 59 | \( 1 - 8.19T + 59T^{2} \) |
| 61 | \( 1 - 8.12T + 61T^{2} \) |
| 67 | \( 1 + 3.24T + 67T^{2} \) |
| 71 | \( 1 + 4.39T + 71T^{2} \) |
| 73 | \( 1 + 14.8T + 73T^{2} \) |
| 79 | \( 1 - 5.71T + 79T^{2} \) |
| 83 | \( 1 + 0.464T + 83T^{2} \) |
| 89 | \( 1 - 1.93T + 89T^{2} \) |
| 97 | \( 1 - 1.69T + 97T^{2} \) |
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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
−8.118073184817637714855495027964, −7.56946156441224421361118467895, −7.11199294391534707543111675144, −5.57549520861760620719906792342, −4.86108923662591377070085841221, −4.09699120139499603791842881016, −3.37830137263234126847046441831, −2.48618782955381324888580235419, −1.74708163067258245097425488924, 0,
1.74708163067258245097425488924, 2.48618782955381324888580235419, 3.37830137263234126847046441831, 4.09699120139499603791842881016, 4.86108923662591377070085841221, 5.57549520861760620719906792342, 7.11199294391534707543111675144, 7.56946156441224421361118467895, 8.118073184817637714855495027964
