| L(s) = 1 | − 2.75·2-s + 3.04·3-s + 5.61·4-s − 2.57·5-s − 8.40·6-s − 0.275·7-s − 9.96·8-s + 6.29·9-s + 7.09·10-s + 1.07·11-s + 17.1·12-s + 3.84·13-s + 0.761·14-s − 7.84·15-s + 16.2·16-s + 17-s − 17.3·18-s − 8.39·19-s − 14.4·20-s − 0.840·21-s − 2.97·22-s − 5.83·23-s − 30.3·24-s + 1.61·25-s − 10.6·26-s + 10.0·27-s − 1.54·28-s + ⋯ |
| L(s) = 1 | − 1.95·2-s + 1.75·3-s + 2.80·4-s − 1.15·5-s − 3.43·6-s − 0.104·7-s − 3.52·8-s + 2.09·9-s + 2.24·10-s + 0.324·11-s + 4.93·12-s + 1.06·13-s + 0.203·14-s − 2.02·15-s + 4.06·16-s + 0.242·17-s − 4.09·18-s − 1.92·19-s − 3.22·20-s − 0.183·21-s − 0.633·22-s − 1.21·23-s − 6.20·24-s + 0.323·25-s − 2.08·26-s + 1.92·27-s − 0.292·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 17 | \( 1 - T \) |
| 353 | \( 1 - T \) |
| good | 2 | \( 1 + 2.75T + 2T^{2} \) |
| 3 | \( 1 - 3.04T + 3T^{2} \) |
| 5 | \( 1 + 2.57T + 5T^{2} \) |
| 7 | \( 1 + 0.275T + 7T^{2} \) |
| 11 | \( 1 - 1.07T + 11T^{2} \) |
| 13 | \( 1 - 3.84T + 13T^{2} \) |
| 19 | \( 1 + 8.39T + 19T^{2} \) |
| 23 | \( 1 + 5.83T + 23T^{2} \) |
| 29 | \( 1 - 9.83T + 29T^{2} \) |
| 31 | \( 1 + 0.683T + 31T^{2} \) |
| 37 | \( 1 - 2.82T + 37T^{2} \) |
| 41 | \( 1 + 11.8T + 41T^{2} \) |
| 43 | \( 1 - 1.63T + 43T^{2} \) |
| 47 | \( 1 + 8.40T + 47T^{2} \) |
| 53 | \( 1 + 8.09T + 53T^{2} \) |
| 59 | \( 1 + 4.19T + 59T^{2} \) |
| 61 | \( 1 - 7.21T + 61T^{2} \) |
| 67 | \( 1 + 3.59T + 67T^{2} \) |
| 71 | \( 1 + 6.33T + 71T^{2} \) |
| 73 | \( 1 - 0.910T + 73T^{2} \) |
| 79 | \( 1 + 4.28T + 79T^{2} \) |
| 83 | \( 1 + 4.00T + 83T^{2} \) |
| 89 | \( 1 - 12.2T + 89T^{2} \) |
| 97 | \( 1 - 15.4T + 97T^{2} \) |
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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
−8.110613888388655594605582661034, −7.55887667637884223073029161459, −6.59619237209581657849494605318, −6.33998991856175383291701081446, −4.39742918982808930463443763515, −3.60046056176037396635033544937, −3.03025659034174887858332648719, −2.07404291745951664769015706797, −1.37807654432260641955995030222, 0,
1.37807654432260641955995030222, 2.07404291745951664769015706797, 3.03025659034174887858332648719, 3.60046056176037396635033544937, 4.39742918982808930463443763515, 6.33998991856175383291701081446, 6.59619237209581657849494605318, 7.55887667637884223073029161459, 8.110613888388655594605582661034
