| L(s) = 1 | − 2.76·2-s − 1.93·3-s + 5.62·4-s − 0.205·5-s + 5.35·6-s − 1.99·7-s − 10.0·8-s + 0.761·9-s + 0.567·10-s − 3.85·11-s − 10.9·12-s + 2.35·13-s + 5.49·14-s + 0.398·15-s + 16.3·16-s + 6.01·17-s − 2.10·18-s − 0.129·19-s − 1.15·20-s + 3.86·21-s + 10.6·22-s + 3.90·23-s + 19.4·24-s − 4.95·25-s − 6.49·26-s + 4.34·27-s − 11.1·28-s + ⋯ |
| L(s) = 1 | − 1.95·2-s − 1.11·3-s + 2.81·4-s − 0.0918·5-s + 2.18·6-s − 0.752·7-s − 3.53·8-s + 0.253·9-s + 0.179·10-s − 1.16·11-s − 3.14·12-s + 0.652·13-s + 1.46·14-s + 0.102·15-s + 4.09·16-s + 1.45·17-s − 0.495·18-s − 0.0297·19-s − 0.258·20-s + 0.842·21-s + 2.27·22-s + 0.815·23-s + 3.96·24-s − 0.991·25-s − 1.27·26-s + 0.835·27-s − 2.11·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 53 | \( 1 + T \) |
| 151 | \( 1 + T \) |
| good | 2 | \( 1 + 2.76T + 2T^{2} \) |
| 3 | \( 1 + 1.93T + 3T^{2} \) |
| 5 | \( 1 + 0.205T + 5T^{2} \) |
| 7 | \( 1 + 1.99T + 7T^{2} \) |
| 11 | \( 1 + 3.85T + 11T^{2} \) |
| 13 | \( 1 - 2.35T + 13T^{2} \) |
| 17 | \( 1 - 6.01T + 17T^{2} \) |
| 19 | \( 1 + 0.129T + 19T^{2} \) |
| 23 | \( 1 - 3.90T + 23T^{2} \) |
| 29 | \( 1 - 0.957T + 29T^{2} \) |
| 31 | \( 1 + 1.15T + 31T^{2} \) |
| 37 | \( 1 + 7.73T + 37T^{2} \) |
| 41 | \( 1 + 4.49T + 41T^{2} \) |
| 43 | \( 1 + 0.734T + 43T^{2} \) |
| 47 | \( 1 - 0.262T + 47T^{2} \) |
| 59 | \( 1 - 11.0T + 59T^{2} \) |
| 61 | \( 1 + 3.50T + 61T^{2} \) |
| 67 | \( 1 + 14.4T + 67T^{2} \) |
| 71 | \( 1 - 14.4T + 71T^{2} \) |
| 73 | \( 1 + 11.8T + 73T^{2} \) |
| 79 | \( 1 - 8.94T + 79T^{2} \) |
| 83 | \( 1 + 9.37T + 83T^{2} \) |
| 89 | \( 1 - 5.83T + 89T^{2} \) |
| 97 | \( 1 + 8.77T + 97T^{2} \) |
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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
−7.50954557233464691590016205733, −7.01203700141808726359322497303, −6.22895586788366074227986868370, −5.75268137562569443151002385044, −5.12473828452309234040668024952, −3.44981812381664460000936066572, −2.92761660669760521010088145362, −1.77846501893704538905815844187, −0.789480232938995832854427056645, 0,
0.789480232938995832854427056645, 1.77846501893704538905815844187, 2.92761660669760521010088145362, 3.44981812381664460000936066572, 5.12473828452309234040668024952, 5.75268137562569443151002385044, 6.22895586788366074227986868370, 7.01203700141808726359322497303, 7.50954557233464691590016205733
