| L(s) = 1 | + 2-s − 3-s + 4-s − 0.0471·5-s − 6-s − 3.98·7-s + 8-s + 9-s − 0.0471·10-s − 5.70·11-s − 12-s − 2.39·13-s − 3.98·14-s + 0.0471·15-s + 16-s + 1.20·17-s + 18-s − 3.98·19-s − 0.0471·20-s + 3.98·21-s − 5.70·22-s + 23-s − 24-s − 4.99·25-s − 2.39·26-s − 27-s − 3.98·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.0210·5-s − 0.408·6-s − 1.50·7-s + 0.353·8-s + 0.333·9-s − 0.0149·10-s − 1.71·11-s − 0.288·12-s − 0.665·13-s − 1.06·14-s + 0.0121·15-s + 0.250·16-s + 0.291·17-s + 0.235·18-s − 0.913·19-s − 0.0105·20-s + 0.869·21-s − 1.21·22-s + 0.208·23-s − 0.204·24-s − 0.999·25-s − 0.470·26-s − 0.192·27-s − 0.752·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.264313211\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.264313211\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 5 | \( 1 + 0.0471T + 5T^{2} \) |
| 7 | \( 1 + 3.98T + 7T^{2} \) |
| 11 | \( 1 + 5.70T + 11T^{2} \) |
| 13 | \( 1 + 2.39T + 13T^{2} \) |
| 17 | \( 1 - 1.20T + 17T^{2} \) |
| 19 | \( 1 + 3.98T + 19T^{2} \) |
| 31 | \( 1 - 7.22T + 31T^{2} \) |
| 37 | \( 1 - 6.50T + 37T^{2} \) |
| 41 | \( 1 - 7.53T + 41T^{2} \) |
| 43 | \( 1 - 9.23T + 43T^{2} \) |
| 47 | \( 1 - 5.96T + 47T^{2} \) |
| 53 | \( 1 - 5.81T + 53T^{2} \) |
| 59 | \( 1 - 4.24T + 59T^{2} \) |
| 61 | \( 1 - 0.600T + 61T^{2} \) |
| 67 | \( 1 + 8.33T + 67T^{2} \) |
| 71 | \( 1 - 1.23T + 71T^{2} \) |
| 73 | \( 1 - 5.54T + 73T^{2} \) |
| 79 | \( 1 + 11.1T + 79T^{2} \) |
| 83 | \( 1 - 3.18T + 83T^{2} \) |
| 89 | \( 1 + 11.3T + 89T^{2} \) |
| 97 | \( 1 + 8.49T + 97T^{2} \) |
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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
−8.237961655990630483006800877139, −7.50051970224511525974045979271, −6.88243932304139205744709405718, −5.90497917905464754934484115952, −5.74907081881693108670863331065, −4.65077518510608800618745056330, −4.00515728557549110617779926585, −2.81360199195475935709795310286, −2.44963580199349348031386369385, −0.55676331421238503860253861745,
0.55676331421238503860253861745, 2.44963580199349348031386369385, 2.81360199195475935709795310286, 4.00515728557549110617779926585, 4.65077518510608800618745056330, 5.74907081881693108670863331065, 5.90497917905464754934484115952, 6.88243932304139205744709405718, 7.50051970224511525974045979271, 8.237961655990630483006800877139
