| L(s) = 1 | − 1.13·2-s − 3-s − 0.702·4-s + 4.37·5-s + 1.13·6-s + 2.87·7-s + 3.07·8-s + 9-s − 4.98·10-s − 4.98·11-s + 0.702·12-s + 0.467·13-s − 3.27·14-s − 4.37·15-s − 2.10·16-s + 17-s − 1.13·18-s + 8.07·19-s − 3.07·20-s − 2.87·21-s + 5.67·22-s − 9.12·23-s − 3.07·24-s + 14.1·25-s − 0.532·26-s − 27-s − 2.02·28-s + ⋯ |
| L(s) = 1 | − 0.805·2-s − 0.577·3-s − 0.351·4-s + 1.95·5-s + 0.465·6-s + 1.08·7-s + 1.08·8-s + 0.333·9-s − 1.57·10-s − 1.50·11-s + 0.202·12-s + 0.129·13-s − 0.876·14-s − 1.12·15-s − 0.525·16-s + 0.242·17-s − 0.268·18-s + 1.85·19-s − 0.687·20-s − 0.628·21-s + 1.21·22-s − 1.90·23-s − 0.628·24-s + 2.82·25-s − 0.104·26-s − 0.192·27-s − 0.382·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.527164764\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.527164764\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 157 | \( 1 + T \) |
| good | 2 | \( 1 + 1.13T + 2T^{2} \) |
| 5 | \( 1 - 4.37T + 5T^{2} \) |
| 7 | \( 1 - 2.87T + 7T^{2} \) |
| 11 | \( 1 + 4.98T + 11T^{2} \) |
| 13 | \( 1 - 0.467T + 13T^{2} \) |
| 19 | \( 1 - 8.07T + 19T^{2} \) |
| 23 | \( 1 + 9.12T + 23T^{2} \) |
| 29 | \( 1 + 3.46T + 29T^{2} \) |
| 31 | \( 1 - 1.58T + 31T^{2} \) |
| 37 | \( 1 - 2.71T + 37T^{2} \) |
| 41 | \( 1 + 12.3T + 41T^{2} \) |
| 43 | \( 1 + 5.71T + 43T^{2} \) |
| 47 | \( 1 + 5.46T + 47T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 + 6.45T + 59T^{2} \) |
| 61 | \( 1 - 4.32T + 61T^{2} \) |
| 67 | \( 1 - 9.87T + 67T^{2} \) |
| 71 | \( 1 + 1.10T + 71T^{2} \) |
| 73 | \( 1 - 1.32T + 73T^{2} \) |
| 79 | \( 1 - 11.7T + 79T^{2} \) |
| 83 | \( 1 - 11.3T + 83T^{2} \) |
| 89 | \( 1 - 12.0T + 89T^{2} \) |
| 97 | \( 1 - 5.38T + 97T^{2} \) |
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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
−7.965843720473543052641612036183, −7.33007923635159243173017116061, −6.37130585328885969064357763846, −5.52846529557595921707926869934, −5.20333483709505541485678899717, −4.77563381251049759991192014376, −3.40579273987531797042520173776, −2.12483376071528389600351914538, −1.72555908815549091156293627577, −0.74560501276981968248926359965,
0.74560501276981968248926359965, 1.72555908815549091156293627577, 2.12483376071528389600351914538, 3.40579273987531797042520173776, 4.77563381251049759991192014376, 5.20333483709505541485678899717, 5.52846529557595921707926869934, 6.37130585328885969064357763846, 7.33007923635159243173017116061, 7.965843720473543052641612036183
