| L(s) = 1 | − 1.99·2-s − 2.94·3-s + 1.96·4-s + 5.85·6-s − 3.47·7-s + 0.0709·8-s + 5.65·9-s − 3.24·11-s − 5.77·12-s + 3.53·13-s + 6.92·14-s − 4.07·16-s − 11.2·18-s + 1.42·19-s + 10.2·21-s + 6.46·22-s + 7.40·23-s − 0.208·24-s − 7.03·26-s − 7.79·27-s − 6.83·28-s − 10.4·29-s − 1.74·31-s + 7.96·32-s + 9.54·33-s + 11.0·36-s − 3.36·37-s + ⋯ |
| L(s) = 1 | − 1.40·2-s − 1.69·3-s + 0.982·4-s + 2.39·6-s − 1.31·7-s + 0.0251·8-s + 1.88·9-s − 0.978·11-s − 1.66·12-s + 0.979·13-s + 1.85·14-s − 1.01·16-s − 2.65·18-s + 0.326·19-s + 2.23·21-s + 1.37·22-s + 1.54·23-s − 0.0426·24-s − 1.37·26-s − 1.50·27-s − 1.29·28-s − 1.93·29-s − 0.313·31-s + 1.40·32-s + 1.66·33-s + 1.84·36-s − 0.553·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + 1.99T + 2T^{2} \) |
| 3 | \( 1 + 2.94T + 3T^{2} \) |
| 7 | \( 1 + 3.47T + 7T^{2} \) |
| 11 | \( 1 + 3.24T + 11T^{2} \) |
| 13 | \( 1 - 3.53T + 13T^{2} \) |
| 19 | \( 1 - 1.42T + 19T^{2} \) |
| 23 | \( 1 - 7.40T + 23T^{2} \) |
| 29 | \( 1 + 10.4T + 29T^{2} \) |
| 31 | \( 1 + 1.74T + 31T^{2} \) |
| 37 | \( 1 + 3.36T + 37T^{2} \) |
| 41 | \( 1 - 3.91T + 41T^{2} \) |
| 43 | \( 1 + 6.68T + 43T^{2} \) |
| 47 | \( 1 + 1.75T + 47T^{2} \) |
| 53 | \( 1 + 2.10T + 53T^{2} \) |
| 59 | \( 1 - 6.95T + 59T^{2} \) |
| 61 | \( 1 - 7.31T + 61T^{2} \) |
| 67 | \( 1 + 0.934T + 67T^{2} \) |
| 71 | \( 1 + 16.5T + 71T^{2} \) |
| 73 | \( 1 - 2.58T + 73T^{2} \) |
| 79 | \( 1 - 11.6T + 79T^{2} \) |
| 83 | \( 1 + 6.23T + 83T^{2} \) |
| 89 | \( 1 - 10.6T + 89T^{2} \) |
| 97 | \( 1 + 4.47T + 97T^{2} \) |
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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
−7.40836021832427240498702933003, −6.94563379591907070253413930420, −6.36022327203892939932869475562, −5.57844861588883252675943559444, −5.08369525041850112098523278722, −3.97700771719244411000965456597, −3.03735315044903991897585737769, −1.72574691163623699570083992809, −0.74643705496915680529712658322, 0,
0.74643705496915680529712658322, 1.72574691163623699570083992809, 3.03735315044903991897585737769, 3.97700771719244411000965456597, 5.08369525041850112098523278722, 5.57844861588883252675943559444, 6.36022327203892939932869475562, 6.94563379591907070253413930420, 7.40836021832427240498702933003
