| L(s) = 1 | − 2.51·2-s − 2.93·3-s + 4.30·4-s − 3.06·5-s + 7.36·6-s + 3.82·7-s − 5.78·8-s + 5.60·9-s + 7.70·10-s − 0.749·11-s − 12.6·12-s + 13-s − 9.59·14-s + 8.99·15-s + 5.92·16-s − 6.57·17-s − 14.0·18-s + 5.95·19-s − 13.2·20-s − 11.2·21-s + 1.88·22-s + 7.17·23-s + 16.9·24-s + 4.40·25-s − 2.51·26-s − 7.62·27-s + 16.4·28-s + ⋯ |
| L(s) = 1 | − 1.77·2-s − 1.69·3-s + 2.15·4-s − 1.37·5-s + 3.00·6-s + 1.44·7-s − 2.04·8-s + 1.86·9-s + 2.43·10-s − 0.225·11-s − 3.64·12-s + 0.277·13-s − 2.56·14-s + 2.32·15-s + 1.48·16-s − 1.59·17-s − 3.31·18-s + 1.36·19-s − 2.95·20-s − 2.44·21-s + 0.401·22-s + 1.49·23-s + 3.46·24-s + 0.881·25-s − 0.492·26-s − 1.46·27-s + 3.10·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8047 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8047 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 - T \) |
| 619 | \( 1 - T \) |
| good | 2 | \( 1 + 2.51T + 2T^{2} \) |
| 3 | \( 1 + 2.93T + 3T^{2} \) |
| 5 | \( 1 + 3.06T + 5T^{2} \) |
| 7 | \( 1 - 3.82T + 7T^{2} \) |
| 11 | \( 1 + 0.749T + 11T^{2} \) |
| 17 | \( 1 + 6.57T + 17T^{2} \) |
| 19 | \( 1 - 5.95T + 19T^{2} \) |
| 23 | \( 1 - 7.17T + 23T^{2} \) |
| 29 | \( 1 - 0.412T + 29T^{2} \) |
| 31 | \( 1 - 0.475T + 31T^{2} \) |
| 37 | \( 1 - 7.83T + 37T^{2} \) |
| 41 | \( 1 + 6.09T + 41T^{2} \) |
| 43 | \( 1 - 3.56T + 43T^{2} \) |
| 47 | \( 1 + 7.17T + 47T^{2} \) |
| 53 | \( 1 - 11.7T + 53T^{2} \) |
| 59 | \( 1 + 8.83T + 59T^{2} \) |
| 61 | \( 1 + 4.24T + 61T^{2} \) |
| 67 | \( 1 + 11.6T + 67T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 - 2.64T + 73T^{2} \) |
| 79 | \( 1 + 4.45T + 79T^{2} \) |
| 83 | \( 1 + 13.0T + 83T^{2} \) |
| 89 | \( 1 - 18.6T + 89T^{2} \) |
| 97 | \( 1 + 17.0T + 97T^{2} \) |
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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
−7.64436933707594159799756956909, −6.97530099303368132414359068214, −6.50047352627592148014100597240, −5.41692297892231750209921767206, −4.80248557246489138636491313579, −4.15559133479275932930834464640, −2.78420392758345624440255538831, −1.51364025674112298102868766016, −0.895514942517053520897768227950, 0,
0.895514942517053520897768227950, 1.51364025674112298102868766016, 2.78420392758345624440255538831, 4.15559133479275932930834464640, 4.80248557246489138636491313579, 5.41692297892231750209921767206, 6.50047352627592148014100597240, 6.97530099303368132414359068214, 7.64436933707594159799756956909
