| L(s) = 1 | − 0.717·2-s − 1.56·3-s − 1.48·4-s + 1.12·6-s + 2.51·7-s + 2.49·8-s − 0.535·9-s − 5.85·11-s + 2.33·12-s + 0.791·13-s − 1.80·14-s + 1.17·16-s + 0.651·17-s + 0.384·18-s − 3.95·21-s + 4.19·22-s − 4.88·23-s − 3.92·24-s − 0.567·26-s + 5.55·27-s − 3.74·28-s − 4.83·29-s − 6.73·31-s − 5.84·32-s + 9.18·33-s − 0.467·34-s + 0.796·36-s + ⋯ |
| L(s) = 1 | − 0.507·2-s − 0.906·3-s − 0.742·4-s + 0.459·6-s + 0.952·7-s + 0.883·8-s − 0.178·9-s − 1.76·11-s + 0.673·12-s + 0.219·13-s − 0.482·14-s + 0.294·16-s + 0.157·17-s + 0.0905·18-s − 0.862·21-s + 0.895·22-s − 1.01·23-s − 0.800·24-s − 0.111·26-s + 1.06·27-s − 0.707·28-s − 0.897·29-s − 1.21·31-s − 1.03·32-s + 1.59·33-s − 0.0801·34-s + 0.132·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3422294006\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3422294006\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 0.717T + 2T^{2} \) |
| 3 | \( 1 + 1.56T + 3T^{2} \) |
| 7 | \( 1 - 2.51T + 7T^{2} \) |
| 11 | \( 1 + 5.85T + 11T^{2} \) |
| 13 | \( 1 - 0.791T + 13T^{2} \) |
| 17 | \( 1 - 0.651T + 17T^{2} \) |
| 23 | \( 1 + 4.88T + 23T^{2} \) |
| 29 | \( 1 + 4.83T + 29T^{2} \) |
| 31 | \( 1 + 6.73T + 31T^{2} \) |
| 37 | \( 1 + 0.741T + 37T^{2} \) |
| 41 | \( 1 - 8.04T + 41T^{2} \) |
| 43 | \( 1 + 0.761T + 43T^{2} \) |
| 47 | \( 1 - 11.3T + 47T^{2} \) |
| 53 | \( 1 + 12.8T + 53T^{2} \) |
| 59 | \( 1 + 2.14T + 59T^{2} \) |
| 61 | \( 1 + 6.75T + 61T^{2} \) |
| 67 | \( 1 + 13.8T + 67T^{2} \) |
| 71 | \( 1 + 6.05T + 71T^{2} \) |
| 73 | \( 1 - 11.1T + 73T^{2} \) |
| 79 | \( 1 - 15.7T + 79T^{2} \) |
| 83 | \( 1 - 3.26T + 83T^{2} \) |
| 89 | \( 1 - 1.07T + 89T^{2} \) |
| 97 | \( 1 + 10.1T + 97T^{2} \) |
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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
−7.73971642557571602158993676636, −7.43951922814776186490519405040, −6.09699112919943310593461016403, −5.60621602953894915872944966656, −5.03909373492790656561519007483, −4.51957458018743700706173264929, −3.57068172505977932704965761248, −2.43528642088897228481829666532, −1.49205231683194259555609694042, −0.33226775287382097988658120960,
0.33226775287382097988658120960, 1.49205231683194259555609694042, 2.43528642088897228481829666532, 3.57068172505977932704965761248, 4.51957458018743700706173264929, 5.03909373492790656561519007483, 5.60621602953894915872944966656, 6.09699112919943310593461016403, 7.43951922814776186490519405040, 7.73971642557571602158993676636
