| L(s) = 1 | + 2-s + 4-s + 4.30·5-s − 2.58·7-s + 8-s + 4.30·10-s − 2.36·11-s − 4.34·13-s − 2.58·14-s + 16-s − 4.59·17-s + 4.77·19-s + 4.30·20-s − 2.36·22-s + 6.85·23-s + 13.5·25-s − 4.34·26-s − 2.58·28-s + 9.68·29-s + 4.83·31-s + 32-s − 4.59·34-s − 11.1·35-s − 4.41·37-s + 4.77·38-s + 4.30·40-s + 4.47·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.92·5-s − 0.975·7-s + 0.353·8-s + 1.36·10-s − 0.711·11-s − 1.20·13-s − 0.689·14-s + 0.250·16-s − 1.11·17-s + 1.09·19-s + 0.962·20-s − 0.503·22-s + 1.43·23-s + 2.70·25-s − 0.852·26-s − 0.487·28-s + 1.79·29-s + 0.867·31-s + 0.176·32-s − 0.788·34-s − 1.87·35-s − 0.726·37-s + 0.774·38-s + 0.680·40-s + 0.699·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.217786865\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.217786865\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 149 | \( 1 - T \) |
| good | 5 | \( 1 - 4.30T + 5T^{2} \) |
| 7 | \( 1 + 2.58T + 7T^{2} \) |
| 11 | \( 1 + 2.36T + 11T^{2} \) |
| 13 | \( 1 + 4.34T + 13T^{2} \) |
| 17 | \( 1 + 4.59T + 17T^{2} \) |
| 19 | \( 1 - 4.77T + 19T^{2} \) |
| 23 | \( 1 - 6.85T + 23T^{2} \) |
| 29 | \( 1 - 9.68T + 29T^{2} \) |
| 31 | \( 1 - 4.83T + 31T^{2} \) |
| 37 | \( 1 + 4.41T + 37T^{2} \) |
| 41 | \( 1 - 4.47T + 41T^{2} \) |
| 43 | \( 1 + 3.56T + 43T^{2} \) |
| 47 | \( 1 - 6.92T + 47T^{2} \) |
| 53 | \( 1 + 2.33T + 53T^{2} \) |
| 59 | \( 1 - 3.02T + 59T^{2} \) |
| 61 | \( 1 + 1.66T + 61T^{2} \) |
| 67 | \( 1 + 14.9T + 67T^{2} \) |
| 71 | \( 1 + 1.22T + 71T^{2} \) |
| 73 | \( 1 - 16.5T + 73T^{2} \) |
| 79 | \( 1 - 0.926T + 79T^{2} \) |
| 83 | \( 1 - 16.1T + 83T^{2} \) |
| 89 | \( 1 - 2.40T + 89T^{2} \) |
| 97 | \( 1 + 9.91T + 97T^{2} \) |
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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
−7.55289887116766391373607702722, −6.74359230269578471240342931275, −6.50821693250487830819402894456, −5.69248153289667163490557372146, −4.99795177786741903118705440659, −4.69463243749732331560966015781, −3.14403539166430566537580353347, −2.73889312768730614157525108150, −2.12915559041455728620127733349, −0.906711056880926843041331449366,
0.906711056880926843041331449366, 2.12915559041455728620127733349, 2.73889312768730614157525108150, 3.14403539166430566537580353347, 4.69463243749732331560966015781, 4.99795177786741903118705440659, 5.69248153289667163490557372146, 6.50821693250487830819402894456, 6.74359230269578471240342931275, 7.55289887116766391373607702722
