| L(s) = 1 | − 1.88·2-s + 1.56·4-s − 3.37·5-s + 1.01·7-s + 0.818·8-s + 6.37·10-s + 3.49·11-s + 6.75·13-s − 1.92·14-s − 4.67·16-s − 0.401·17-s − 4.47·19-s − 5.29·20-s − 6.60·22-s − 23-s + 6.41·25-s − 12.7·26-s + 1.59·28-s + 29-s − 9.89·31-s + 7.19·32-s + 0.758·34-s − 3.44·35-s + 7.54·37-s + 8.44·38-s − 2.76·40-s − 9.45·41-s + ⋯ |
| L(s) = 1 | − 1.33·2-s + 0.783·4-s − 1.51·5-s + 0.384·7-s + 0.289·8-s + 2.01·10-s + 1.05·11-s + 1.87·13-s − 0.514·14-s − 1.16·16-s − 0.0973·17-s − 1.02·19-s − 1.18·20-s − 1.40·22-s − 0.208·23-s + 1.28·25-s − 2.50·26-s + 0.301·28-s + 0.185·29-s − 1.77·31-s + 1.27·32-s + 0.130·34-s − 0.581·35-s + 1.24·37-s + 1.37·38-s − 0.437·40-s − 1.47·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + 1.88T + 2T^{2} \) |
| 5 | \( 1 + 3.37T + 5T^{2} \) |
| 7 | \( 1 - 1.01T + 7T^{2} \) |
| 11 | \( 1 - 3.49T + 11T^{2} \) |
| 13 | \( 1 - 6.75T + 13T^{2} \) |
| 17 | \( 1 + 0.401T + 17T^{2} \) |
| 19 | \( 1 + 4.47T + 19T^{2} \) |
| 31 | \( 1 + 9.89T + 31T^{2} \) |
| 37 | \( 1 - 7.54T + 37T^{2} \) |
| 41 | \( 1 + 9.45T + 41T^{2} \) |
| 43 | \( 1 + 0.331T + 43T^{2} \) |
| 47 | \( 1 - 3.53T + 47T^{2} \) |
| 53 | \( 1 - 6.65T + 53T^{2} \) |
| 59 | \( 1 + 5.79T + 59T^{2} \) |
| 61 | \( 1 + 2.17T + 61T^{2} \) |
| 67 | \( 1 - 5.30T + 67T^{2} \) |
| 71 | \( 1 + 8.86T + 71T^{2} \) |
| 73 | \( 1 + 2.05T + 73T^{2} \) |
| 79 | \( 1 + 1.29T + 79T^{2} \) |
| 83 | \( 1 - 2.18T + 83T^{2} \) |
| 89 | \( 1 + 14.4T + 89T^{2} \) |
| 97 | \( 1 + 14.3T + 97T^{2} \) |
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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
−8.029810363207713411518318514705, −7.21627988840843311760019631027, −6.67050342256128747101555164895, −5.81959881917252501730404413545, −4.48933042877002733676083989421, −4.04549894033716199813932377464, −3.35845454163420416059863617833, −1.84335419377124193058391927278, −1.09142836567624690355554326350, 0,
1.09142836567624690355554326350, 1.84335419377124193058391927278, 3.35845454163420416059863617833, 4.04549894033716199813932377464, 4.48933042877002733676083989421, 5.81959881917252501730404413545, 6.67050342256128747101555164895, 7.21627988840843311760019631027, 8.029810363207713411518318514705
