| L(s) = 1 | − 0.289·2-s − 1.91·4-s + 0.894·5-s + 3.31·7-s + 1.13·8-s − 0.258·10-s + 3.47·11-s + 1.33·13-s − 0.957·14-s + 3.50·16-s − 4.10·17-s − 2.37·19-s − 1.71·20-s − 1.00·22-s − 23-s − 4.20·25-s − 0.385·26-s − 6.34·28-s − 29-s − 8.16·31-s − 3.27·32-s + 1.18·34-s + 2.96·35-s − 7.23·37-s + 0.686·38-s + 1.01·40-s − 2.74·41-s + ⋯ |
| L(s) = 1 | − 0.204·2-s − 0.958·4-s + 0.399·5-s + 1.25·7-s + 0.400·8-s − 0.0817·10-s + 1.04·11-s + 0.370·13-s − 0.255·14-s + 0.876·16-s − 0.995·17-s − 0.544·19-s − 0.383·20-s − 0.214·22-s − 0.208·23-s − 0.840·25-s − 0.0756·26-s − 1.19·28-s − 0.185·29-s − 1.46·31-s − 0.579·32-s + 0.203·34-s + 0.500·35-s − 1.18·37-s + 0.111·38-s + 0.160·40-s − 0.428·41-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 | \( 1 + 0.289T + 2T^{2} \) |
| 5 | \( 1 - 0.894T + 5T^{2} \) |
| 7 | \( 1 - 3.31T + 7T^{2} \) |
| 11 | \( 1 - 3.47T + 11T^{2} \) |
| 13 | \( 1 - 1.33T + 13T^{2} \) |
| 17 | \( 1 + 4.10T + 17T^{2} \) |
| 19 | \( 1 + 2.37T + 19T^{2} \) |
| 31 | \( 1 + 8.16T + 31T^{2} \) |
| 37 | \( 1 + 7.23T + 37T^{2} \) |
| 41 | \( 1 + 2.74T + 41T^{2} \) |
| 43 | \( 1 - 1.38T + 43T^{2} \) |
| 47 | \( 1 + 2.36T + 47T^{2} \) |
| 53 | \( 1 + 1.52T + 53T^{2} \) |
| 59 | \( 1 + 15.2T + 59T^{2} \) |
| 61 | \( 1 - 4.80T + 61T^{2} \) |
| 67 | \( 1 + 12.6T + 67T^{2} \) |
| 71 | \( 1 - 0.717T + 71T^{2} \) |
| 73 | \( 1 + 11.3T + 73T^{2} \) |
| 79 | \( 1 - 8.33T + 79T^{2} \) |
| 83 | \( 1 - 0.376T + 83T^{2} \) |
| 89 | \( 1 + 17.2T + 89T^{2} \) |
| 97 | \( 1 - 11.7T + 97T^{2} \) |
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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
−7.86137525746194543403619952458, −7.14168063001375193289394410808, −6.20460670970239813931297704032, −5.54229294397433217096181271944, −4.70729469373305420329915556533, −4.20831730212253707145925341309, −3.44020322965040112523580355994, −1.90333331350515447027208614303, −1.49186955519047113250633911969, 0,
1.49186955519047113250633911969, 1.90333331350515447027208614303, 3.44020322965040112523580355994, 4.20831730212253707145925341309, 4.70729469373305420329915556533, 5.54229294397433217096181271944, 6.20460670970239813931297704032, 7.14168063001375193289394410808, 7.86137525746194543403619952458
