| L(s) = 1 | − 0.982·2-s − 2.74·3-s − 1.03·4-s − 0.473·5-s + 2.69·6-s + 0.734·7-s + 2.98·8-s + 4.52·9-s + 0.465·10-s + 11-s + 2.83·12-s + 4.26·13-s − 0.721·14-s + 1.29·15-s − 0.860·16-s + 5.99·17-s − 4.44·18-s + 0.434·19-s + 0.489·20-s − 2.01·21-s − 0.982·22-s − 0.899·23-s − 8.18·24-s − 4.77·25-s − 4.18·26-s − 4.18·27-s − 0.759·28-s + ⋯ |
| L(s) = 1 | − 0.694·2-s − 1.58·3-s − 0.517·4-s − 0.211·5-s + 1.10·6-s + 0.277·7-s + 1.05·8-s + 1.50·9-s + 0.147·10-s + 0.301·11-s + 0.819·12-s + 1.18·13-s − 0.192·14-s + 0.335·15-s − 0.215·16-s + 1.45·17-s − 1.04·18-s + 0.0997·19-s + 0.109·20-s − 0.439·21-s − 0.209·22-s − 0.187·23-s − 1.66·24-s − 0.955·25-s − 0.821·26-s − 0.806·27-s − 0.143·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9251 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9251 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 - T \) |
| 29 | \( 1 \) |
| good | 2 | \( 1 + 0.982T + 2T^{2} \) |
| 3 | \( 1 + 2.74T + 3T^{2} \) |
| 5 | \( 1 + 0.473T + 5T^{2} \) |
| 7 | \( 1 - 0.734T + 7T^{2} \) |
| 13 | \( 1 - 4.26T + 13T^{2} \) |
| 17 | \( 1 - 5.99T + 17T^{2} \) |
| 19 | \( 1 - 0.434T + 19T^{2} \) |
| 23 | \( 1 + 0.899T + 23T^{2} \) |
| 31 | \( 1 - 8.85T + 31T^{2} \) |
| 37 | \( 1 - 6.47T + 37T^{2} \) |
| 41 | \( 1 + 2.26T + 41T^{2} \) |
| 43 | \( 1 - 0.296T + 43T^{2} \) |
| 47 | \( 1 + 0.724T + 47T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 + 5.23T + 59T^{2} \) |
| 61 | \( 1 - 7.03T + 61T^{2} \) |
| 67 | \( 1 + 8.99T + 67T^{2} \) |
| 71 | \( 1 - 2.22T + 71T^{2} \) |
| 73 | \( 1 + 16.9T + 73T^{2} \) |
| 79 | \( 1 + 10.6T + 79T^{2} \) |
| 83 | \( 1 - 11.6T + 83T^{2} \) |
| 89 | \( 1 + 8.40T + 89T^{2} \) |
| 97 | \( 1 + 11.2T + 97T^{2} \) |
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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
−7.51632016247171362859316449141, −6.55997140533829636196000554497, −5.98417492522062035539142515110, −5.43946911354216081539802466266, −4.61251861602774632611829687280, −4.13774470683384198344069224522, −3.18319224208754261461106627363, −1.47500077223886141645448570123, −1.06200735004592964059105603955, 0,
1.06200735004592964059105603955, 1.47500077223886141645448570123, 3.18319224208754261461106627363, 4.13774470683384198344069224522, 4.61251861602774632611829687280, 5.43946911354216081539802466266, 5.98417492522062035539142515110, 6.55997140533829636196000554497, 7.51632016247171362859316449141
