| L(s) = 1 | + 0.958·2-s + 0.899·3-s − 1.08·4-s + 0.862·6-s + 0.761·7-s − 2.95·8-s − 2.19·9-s − 0.971·12-s + 3.25·13-s + 0.730·14-s − 0.670·16-s + 0.782·17-s − 2.10·18-s − 4.59·19-s + 0.684·21-s + 2.51·23-s − 2.65·24-s + 3.12·26-s − 4.66·27-s − 0.823·28-s − 3.37·29-s − 4.90·31-s + 5.26·32-s + 0.750·34-s + 2.36·36-s − 5.69·37-s − 4.40·38-s + ⋯ |
| L(s) = 1 | + 0.677·2-s + 0.519·3-s − 0.540·4-s + 0.352·6-s + 0.287·7-s − 1.04·8-s − 0.730·9-s − 0.280·12-s + 0.903·13-s + 0.195·14-s − 0.167·16-s + 0.189·17-s − 0.495·18-s − 1.05·19-s + 0.149·21-s + 0.524·23-s − 0.542·24-s + 0.612·26-s − 0.898·27-s − 0.155·28-s − 0.627·29-s − 0.880·31-s + 0.930·32-s + 0.128·34-s + 0.394·36-s − 0.935·37-s − 0.715·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 0.958T + 2T^{2} \) |
| 3 | \( 1 - 0.899T + 3T^{2} \) |
| 7 | \( 1 - 0.761T + 7T^{2} \) |
| 13 | \( 1 - 3.25T + 13T^{2} \) |
| 17 | \( 1 - 0.782T + 17T^{2} \) |
| 19 | \( 1 + 4.59T + 19T^{2} \) |
| 23 | \( 1 - 2.51T + 23T^{2} \) |
| 29 | \( 1 + 3.37T + 29T^{2} \) |
| 31 | \( 1 + 4.90T + 31T^{2} \) |
| 37 | \( 1 + 5.69T + 37T^{2} \) |
| 41 | \( 1 - 5.44T + 41T^{2} \) |
| 43 | \( 1 + 10.5T + 43T^{2} \) |
| 47 | \( 1 + 3.00T + 47T^{2} \) |
| 53 | \( 1 - 13.8T + 53T^{2} \) |
| 59 | \( 1 + 14.2T + 59T^{2} \) |
| 61 | \( 1 + 9.44T + 61T^{2} \) |
| 67 | \( 1 - 11.8T + 67T^{2} \) |
| 71 | \( 1 + 2.18T + 71T^{2} \) |
| 73 | \( 1 + 14.4T + 73T^{2} \) |
| 79 | \( 1 + 7.34T + 79T^{2} \) |
| 83 | \( 1 + 0.788T + 83T^{2} \) |
| 89 | \( 1 - 2.41T + 89T^{2} \) |
| 97 | \( 1 + 1.80T + 97T^{2} \) |
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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
−8.555930469948060640001030011119, −7.76719950335542441544829531062, −6.67429716739709602727935645474, −5.85912686169262099115667631471, −5.27074592981229695611833591949, −4.33044756017803161890719432896, −3.58456935454772088308238075874, −2.90666437496449135383942053455, −1.68489163279848589975975089167, 0,
1.68489163279848589975975089167, 2.90666437496449135383942053455, 3.58456935454772088308238075874, 4.33044756017803161890719432896, 5.27074592981229695611833591949, 5.85912686169262099115667631471, 6.67429716739709602727935645474, 7.76719950335542441544829531062, 8.555930469948060640001030011119
