| L(s) = 1 | − 1.23·2-s − 1.22·3-s − 0.476·4-s + 5-s + 1.50·6-s + 7-s + 3.05·8-s − 1.51·9-s − 1.23·10-s − 1.65·11-s + 0.581·12-s − 2.45·13-s − 1.23·14-s − 1.22·15-s − 2.81·16-s + 0.763·17-s + 1.86·18-s + 2.23·19-s − 0.476·20-s − 1.22·21-s + 2.04·22-s − 4.67·23-s − 3.73·24-s + 25-s + 3.03·26-s + 5.50·27-s − 0.476·28-s + ⋯ |
| L(s) = 1 | − 0.872·2-s − 0.704·3-s − 0.238·4-s + 0.447·5-s + 0.614·6-s + 0.377·7-s + 1.08·8-s − 0.503·9-s − 0.390·10-s − 0.499·11-s + 0.167·12-s − 0.681·13-s − 0.329·14-s − 0.315·15-s − 0.704·16-s + 0.185·17-s + 0.439·18-s + 0.512·19-s − 0.106·20-s − 0.266·21-s + 0.435·22-s − 0.975·23-s − 0.761·24-s + 0.200·25-s + 0.594·26-s + 1.05·27-s − 0.0900·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 229 | \( 1 + T \) |
| good | 2 | \( 1 + 1.23T + 2T^{2} \) |
| 3 | \( 1 + 1.22T + 3T^{2} \) |
| 11 | \( 1 + 1.65T + 11T^{2} \) |
| 13 | \( 1 + 2.45T + 13T^{2} \) |
| 17 | \( 1 - 0.763T + 17T^{2} \) |
| 19 | \( 1 - 2.23T + 19T^{2} \) |
| 23 | \( 1 + 4.67T + 23T^{2} \) |
| 29 | \( 1 + 1.38T + 29T^{2} \) |
| 31 | \( 1 - 5.37T + 31T^{2} \) |
| 37 | \( 1 + 3.92T + 37T^{2} \) |
| 41 | \( 1 + 1.63T + 41T^{2} \) |
| 43 | \( 1 + 0.443T + 43T^{2} \) |
| 47 | \( 1 - 7.59T + 47T^{2} \) |
| 53 | \( 1 - 13.3T + 53T^{2} \) |
| 59 | \( 1 - 1.55T + 59T^{2} \) |
| 61 | \( 1 + 4.51T + 61T^{2} \) |
| 67 | \( 1 + 3.92T + 67T^{2} \) |
| 71 | \( 1 + 7.20T + 71T^{2} \) |
| 73 | \( 1 + 9.71T + 73T^{2} \) |
| 79 | \( 1 - 10.6T + 79T^{2} \) |
| 83 | \( 1 + 7.37T + 83T^{2} \) |
| 89 | \( 1 - 3.74T + 89T^{2} \) |
| 97 | \( 1 - 4.83T + 97T^{2} \) |
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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
−7.56424464994216226164336103730, −6.99228154183145510717703622744, −5.97892042251840489242560534049, −5.43795616203829722680443843704, −4.85484749375138359545840828470, −4.08842517581536209463134370707, −2.87697785628123906191545879194, −2.00448865828660497764163363396, −0.959484816743154366743625068356, 0,
0.959484816743154366743625068356, 2.00448865828660497764163363396, 2.87697785628123906191545879194, 4.08842517581536209463134370707, 4.85484749375138359545840828470, 5.43795616203829722680443843704, 5.97892042251840489242560534049, 6.99228154183145510717703622744, 7.56424464994216226164336103730
