| L(s) = 1 | − 2.28·2-s − 0.648·3-s + 3.22·4-s − 1.19·5-s + 1.48·6-s + 3.55·7-s − 2.78·8-s − 2.57·9-s + 2.72·10-s + 11-s − 2.08·12-s + 2.20·13-s − 8.11·14-s + 0.772·15-s − 0.0715·16-s − 17-s + 5.89·18-s − 2.00·19-s − 3.83·20-s − 2.30·21-s − 2.28·22-s + 3.40·23-s + 1.80·24-s − 3.58·25-s − 5.04·26-s + 3.61·27-s + 11.4·28-s + ⋯ |
| L(s) = 1 | − 1.61·2-s − 0.374·3-s + 1.61·4-s − 0.532·5-s + 0.604·6-s + 1.34·7-s − 0.985·8-s − 0.859·9-s + 0.860·10-s + 0.301·11-s − 0.602·12-s + 0.612·13-s − 2.16·14-s + 0.199·15-s − 0.0178·16-s − 0.242·17-s + 1.38·18-s − 0.460·19-s − 0.857·20-s − 0.502·21-s − 0.487·22-s + 0.710·23-s + 0.368·24-s − 0.716·25-s − 0.990·26-s + 0.696·27-s + 2.16·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6555676035\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6555676035\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| good | 2 | \( 1 + 2.28T + 2T^{2} \) |
| 3 | \( 1 + 0.648T + 3T^{2} \) |
| 5 | \( 1 + 1.19T + 5T^{2} \) |
| 7 | \( 1 - 3.55T + 7T^{2} \) |
| 13 | \( 1 - 2.20T + 13T^{2} \) |
| 19 | \( 1 + 2.00T + 19T^{2} \) |
| 23 | \( 1 - 3.40T + 23T^{2} \) |
| 29 | \( 1 - 7.34T + 29T^{2} \) |
| 31 | \( 1 + 2.81T + 31T^{2} \) |
| 37 | \( 1 + 6.89T + 37T^{2} \) |
| 41 | \( 1 - 2.63T + 41T^{2} \) |
| 47 | \( 1 + 0.309T + 47T^{2} \) |
| 53 | \( 1 - 3.94T + 53T^{2} \) |
| 59 | \( 1 + 11.6T + 59T^{2} \) |
| 61 | \( 1 + 6.80T + 61T^{2} \) |
| 67 | \( 1 - 1.66T + 67T^{2} \) |
| 71 | \( 1 - 5.91T + 71T^{2} \) |
| 73 | \( 1 - 7.52T + 73T^{2} \) |
| 79 | \( 1 - 7.35T + 79T^{2} \) |
| 83 | \( 1 - 8.75T + 83T^{2} \) |
| 89 | \( 1 + 13.9T + 89T^{2} \) |
| 97 | \( 1 + 12.7T + 97T^{2} \) |
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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
−8.086349270462854817882526758526, −7.40721402434580912132699457220, −6.67563547124262962009715444656, −5.99462908613096229156485471954, −5.04930240451249612524370222106, −4.40170908399370975816743581565, −3.32419384391310999644480321482, −2.25180009662285163858977433936, −1.45764376136723460632858770320, −0.55693109928376495455926462672,
0.55693109928376495455926462672, 1.45764376136723460632858770320, 2.25180009662285163858977433936, 3.32419384391310999644480321482, 4.40170908399370975816743581565, 5.04930240451249612524370222106, 5.99462908613096229156485471954, 6.67563547124262962009715444656, 7.40721402434580912132699457220, 8.086349270462854817882526758526
