| L(s) = 1 | − 2.52·2-s + 3-s + 4.39·4-s + 0.815·5-s − 2.52·6-s + 5.25·7-s − 6.06·8-s + 9-s − 2.06·10-s − 6.35·11-s + 4.39·12-s − 5.16·13-s − 13.2·14-s + 0.815·15-s + 6.53·16-s − 6.61·17-s − 2.52·18-s + 4.45·19-s + 3.58·20-s + 5.25·21-s + 16.0·22-s + 5.17·23-s − 6.06·24-s − 4.33·25-s + 13.0·26-s + 27-s + 23.0·28-s + ⋯ |
| L(s) = 1 | − 1.78·2-s + 0.577·3-s + 2.19·4-s + 0.364·5-s − 1.03·6-s + 1.98·7-s − 2.14·8-s + 0.333·9-s − 0.652·10-s − 1.91·11-s + 1.26·12-s − 1.43·13-s − 3.55·14-s + 0.210·15-s + 1.63·16-s − 1.60·17-s − 0.596·18-s + 1.02·19-s + 0.801·20-s + 1.14·21-s + 3.42·22-s + 1.07·23-s − 1.23·24-s − 0.866·25-s + 2.56·26-s + 0.192·27-s + 4.36·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.092807859\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.092807859\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 2671 | \( 1 + T \) |
| good | 2 | \( 1 + 2.52T + 2T^{2} \) |
| 5 | \( 1 - 0.815T + 5T^{2} \) |
| 7 | \( 1 - 5.25T + 7T^{2} \) |
| 11 | \( 1 + 6.35T + 11T^{2} \) |
| 13 | \( 1 + 5.16T + 13T^{2} \) |
| 17 | \( 1 + 6.61T + 17T^{2} \) |
| 19 | \( 1 - 4.45T + 19T^{2} \) |
| 23 | \( 1 - 5.17T + 23T^{2} \) |
| 29 | \( 1 + 4.11T + 29T^{2} \) |
| 31 | \( 1 - 3.82T + 31T^{2} \) |
| 37 | \( 1 + 0.123T + 37T^{2} \) |
| 41 | \( 1 - 3.24T + 41T^{2} \) |
| 43 | \( 1 - 1.37T + 43T^{2} \) |
| 47 | \( 1 + 2.44T + 47T^{2} \) |
| 53 | \( 1 - 8.92T + 53T^{2} \) |
| 59 | \( 1 - 1.43T + 59T^{2} \) |
| 61 | \( 1 - 0.278T + 61T^{2} \) |
| 67 | \( 1 - 4.63T + 67T^{2} \) |
| 71 | \( 1 + 2.14T + 71T^{2} \) |
| 73 | \( 1 + 9.87T + 73T^{2} \) |
| 79 | \( 1 + 4.60T + 79T^{2} \) |
| 83 | \( 1 - 4.10T + 83T^{2} \) |
| 89 | \( 1 + 14.0T + 89T^{2} \) |
| 97 | \( 1 - 10.9T + 97T^{2} \) |
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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
−7.897593495428763296116349416462, −7.33736199188489858392280077104, −7.23286598522482148241810996074, −5.79182420104452535559266896359, −5.03852344999499358052135927533, −4.52314818286746545360214392179, −2.83964819746578496511620640061, −2.25494132755679122590651471477, −1.85829634529181045723684273813, −0.63919147992421385834292329491,
0.63919147992421385834292329491, 1.85829634529181045723684273813, 2.25494132755679122590651471477, 2.83964819746578496511620640061, 4.52314818286746545360214392179, 5.03852344999499358052135927533, 5.79182420104452535559266896359, 7.23286598522482148241810996074, 7.33736199188489858392280077104, 7.897593495428763296116349416462
