| L(s) = 1 | − 2.69·2-s − 2.18·3-s + 5.26·4-s − 2.45·5-s + 5.89·6-s − 1.76·7-s − 8.79·8-s + 1.78·9-s + 6.60·10-s + 2.16·11-s − 11.5·12-s + 3.56·13-s + 4.75·14-s + 5.36·15-s + 13.1·16-s + 6.91·17-s − 4.80·18-s − 6.09·19-s − 12.9·20-s + 3.85·21-s − 5.82·22-s + 4.86·23-s + 19.2·24-s + 1.00·25-s − 9.60·26-s + 2.65·27-s − 9.28·28-s + ⋯ |
| L(s) = 1 | − 1.90·2-s − 1.26·3-s + 2.63·4-s − 1.09·5-s + 2.40·6-s − 0.666·7-s − 3.11·8-s + 0.594·9-s + 2.08·10-s + 0.651·11-s − 3.32·12-s + 0.988·13-s + 1.27·14-s + 1.38·15-s + 3.29·16-s + 1.67·17-s − 1.13·18-s − 1.39·19-s − 2.88·20-s + 0.842·21-s − 1.24·22-s + 1.01·23-s + 3.92·24-s + 0.201·25-s − 1.88·26-s + 0.511·27-s − 1.75·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 6029 | \( 1+O(T) \) |
| good | 2 | \( 1 + 2.69T + 2T^{2} \) |
| 3 | \( 1 + 2.18T + 3T^{2} \) |
| 5 | \( 1 + 2.45T + 5T^{2} \) |
| 7 | \( 1 + 1.76T + 7T^{2} \) |
| 11 | \( 1 - 2.16T + 11T^{2} \) |
| 13 | \( 1 - 3.56T + 13T^{2} \) |
| 17 | \( 1 - 6.91T + 17T^{2} \) |
| 19 | \( 1 + 6.09T + 19T^{2} \) |
| 23 | \( 1 - 4.86T + 23T^{2} \) |
| 29 | \( 1 + 7.34T + 29T^{2} \) |
| 31 | \( 1 - 4.06T + 31T^{2} \) |
| 37 | \( 1 + 5.33T + 37T^{2} \) |
| 41 | \( 1 + 8.06T + 41T^{2} \) |
| 43 | \( 1 + 10.1T + 43T^{2} \) |
| 47 | \( 1 - 1.75T + 47T^{2} \) |
| 53 | \( 1 + 0.144T + 53T^{2} \) |
| 59 | \( 1 - 9.34T + 59T^{2} \) |
| 61 | \( 1 - 0.561T + 61T^{2} \) |
| 67 | \( 1 + 2.85T + 67T^{2} \) |
| 71 | \( 1 - 11.0T + 71T^{2} \) |
| 73 | \( 1 + 8.85T + 73T^{2} \) |
| 79 | \( 1 + 17.0T + 79T^{2} \) |
| 83 | \( 1 - 15.2T + 83T^{2} \) |
| 89 | \( 1 - 11.9T + 89T^{2} \) |
| 97 | \( 1 - 15.7T + 97T^{2} \) |
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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
−7.80479746478804254424216040997, −7.06909457879082748781362007829, −6.51068493582863591972428125549, −6.04190823860956313320218732095, −5.11296504712134492665095451597, −3.72436299476954665387106432314, −3.21622196516051857412048183416, −1.70770037165217801124410196035, −0.813513073978765801466144406377, 0,
0.813513073978765801466144406377, 1.70770037165217801124410196035, 3.21622196516051857412048183416, 3.72436299476954665387106432314, 5.11296504712134492665095451597, 6.04190823860956313320218732095, 6.51068493582863591972428125549, 7.06909457879082748781362007829, 7.80479746478804254424216040997
