| L(s) = 1 | + 2.68·2-s + 5.22·4-s + 0.687·5-s − 3.46·7-s + 8.66·8-s + 1.84·10-s − 4.59·11-s − 5.14·13-s − 9.30·14-s + 12.8·16-s − 5.97·17-s + 2.27·19-s + 3.59·20-s − 12.3·22-s − 23-s − 4.52·25-s − 13.8·26-s − 18.0·28-s − 29-s + 1.71·31-s + 17.1·32-s − 16.0·34-s − 2.38·35-s + 1.76·37-s + 6.12·38-s + 5.95·40-s − 0.217·41-s + ⋯ |
| L(s) = 1 | + 1.90·2-s + 2.61·4-s + 0.307·5-s − 1.30·7-s + 3.06·8-s + 0.584·10-s − 1.38·11-s − 1.42·13-s − 2.48·14-s + 3.21·16-s − 1.44·17-s + 0.523·19-s + 0.803·20-s − 2.63·22-s − 0.208·23-s − 0.905·25-s − 2.71·26-s − 3.41·28-s − 0.185·29-s + 0.308·31-s + 3.03·32-s − 2.75·34-s − 0.402·35-s + 0.289·37-s + 0.994·38-s + 0.942·40-s − 0.0339·41-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 | \( 1 - 2.68T + 2T^{2} \) |
| 5 | \( 1 - 0.687T + 5T^{2} \) |
| 7 | \( 1 + 3.46T + 7T^{2} \) |
| 11 | \( 1 + 4.59T + 11T^{2} \) |
| 13 | \( 1 + 5.14T + 13T^{2} \) |
| 17 | \( 1 + 5.97T + 17T^{2} \) |
| 19 | \( 1 - 2.27T + 19T^{2} \) |
| 31 | \( 1 - 1.71T + 31T^{2} \) |
| 37 | \( 1 - 1.76T + 37T^{2} \) |
| 41 | \( 1 + 0.217T + 41T^{2} \) |
| 43 | \( 1 + 10.2T + 43T^{2} \) |
| 47 | \( 1 - 3.23T + 47T^{2} \) |
| 53 | \( 1 - 9.45T + 53T^{2} \) |
| 59 | \( 1 - 4.74T + 59T^{2} \) |
| 61 | \( 1 - 0.257T + 61T^{2} \) |
| 67 | \( 1 + 12.3T + 67T^{2} \) |
| 71 | \( 1 + 4.10T + 71T^{2} \) |
| 73 | \( 1 - 3.50T + 73T^{2} \) |
| 79 | \( 1 - 12.8T + 79T^{2} \) |
| 83 | \( 1 + 5.50T + 83T^{2} \) |
| 89 | \( 1 + 15.5T + 89T^{2} \) |
| 97 | \( 1 - 18.9T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.27140881591938330778007226195, −6.84980818082949294693508070307, −6.09199351674326872091641621609, −5.47914712916767390244747967882, −4.86731577261561438822078759548, −4.14556773763328028400353820581, −3.23154457533872307063853804092, −2.58715254380091701053735433288, −2.08962073609024995113904104360, 0,
2.08962073609024995113904104360, 2.58715254380091701053735433288, 3.23154457533872307063853804092, 4.14556773763328028400353820581, 4.86731577261561438822078759548, 5.47914712916767390244747967882, 6.09199351674326872091641621609, 6.84980818082949294693508070307, 7.27140881591938330778007226195
