| L(s) = 1 | − 1.34·2-s − 2.34·3-s − 0.195·4-s − 5-s + 3.15·6-s − 7-s + 2.94·8-s + 2.52·9-s + 1.34·10-s + 1.18·11-s + 0.458·12-s − 0.588·13-s + 1.34·14-s + 2.34·15-s − 3.57·16-s + 2.31·17-s − 3.38·18-s − 8.11·19-s + 0.195·20-s + 2.34·21-s − 1.58·22-s − 2.01·23-s − 6.92·24-s + 25-s + 0.790·26-s + 1.12·27-s + 0.195·28-s + ⋯ |
| L(s) = 1 | − 0.949·2-s − 1.35·3-s − 0.0975·4-s − 0.447·5-s + 1.28·6-s − 0.377·7-s + 1.04·8-s + 0.840·9-s + 0.424·10-s + 0.356·11-s + 0.132·12-s − 0.163·13-s + 0.359·14-s + 0.606·15-s − 0.892·16-s + 0.561·17-s − 0.798·18-s − 1.86·19-s + 0.0436·20-s + 0.512·21-s − 0.338·22-s − 0.421·23-s − 1.41·24-s + 0.200·25-s + 0.155·26-s + 0.216·27-s + 0.0368·28-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
| \(L(1)\) |
\(\approx\) |
\(0.3109499403\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3109499403\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.34T + 2T^{2} \) |
| 3 | \( 1 + 2.34T + 3T^{2} \) |
| 11 | \( 1 - 1.18T + 11T^{2} \) |
| 13 | \( 1 + 0.588T + 13T^{2} \) |
| 17 | \( 1 - 2.31T + 17T^{2} \) |
| 19 | \( 1 + 8.11T + 19T^{2} \) |
| 23 | \( 1 + 2.01T + 23T^{2} \) |
| 29 | \( 1 - 9.62T + 29T^{2} \) |
| 31 | \( 1 + 1.07T + 31T^{2} \) |
| 37 | \( 1 - 2.42T + 37T^{2} \) |
| 41 | \( 1 - 5.19T + 41T^{2} \) |
| 43 | \( 1 + 7.17T + 43T^{2} \) |
| 47 | \( 1 - 2.65T + 47T^{2} \) |
| 53 | \( 1 - 3.73T + 53T^{2} \) |
| 59 | \( 1 - 7.26T + 59T^{2} \) |
| 61 | \( 1 + 8.28T + 61T^{2} \) |
| 67 | \( 1 - 2.82T + 67T^{2} \) |
| 71 | \( 1 + 12.9T + 71T^{2} \) |
| 73 | \( 1 - 0.537T + 73T^{2} \) |
| 79 | \( 1 + 6.23T + 79T^{2} \) |
| 83 | \( 1 - 10.6T + 83T^{2} \) |
| 89 | \( 1 - 10.4T + 89T^{2} \) |
| 97 | \( 1 - 11.2T + 97T^{2} \) |
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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
−7.899755556891116574637676477221, −7.13594214788810227887736396524, −6.43837042997303814428380652387, −5.99351884154806991941575782790, −4.93909433206053055343139142701, −4.48758171099736979100469767298, −3.71749549337575960496137184098, −2.43118359123109652108590758603, −1.22892991287225002865215918533, −0.39738392569398358325058006981,
0.39738392569398358325058006981, 1.22892991287225002865215918533, 2.43118359123109652108590758603, 3.71749549337575960496137184098, 4.48758171099736979100469767298, 4.93909433206053055343139142701, 5.99351884154806991941575782790, 6.43837042997303814428380652387, 7.13594214788810227887736396524, 7.899755556891116574637676477221
