| L(s) = 1 | + 2.29·2-s + 3-s + 3.25·4-s + 2.29·6-s − 4.47·7-s + 2.87·8-s + 9-s − 2.51·11-s + 3.25·12-s − 4.43·13-s − 10.2·14-s + 0.0782·16-s − 5.36·17-s + 2.29·18-s + 4.11·19-s − 4.47·21-s − 5.77·22-s − 2.16·23-s + 2.87·24-s − 10.1·26-s + 27-s − 14.5·28-s + 0.758·29-s + 2.74·31-s − 5.56·32-s − 2.51·33-s − 12.3·34-s + ⋯ |
| L(s) = 1 | + 1.62·2-s + 0.577·3-s + 1.62·4-s + 0.935·6-s − 1.69·7-s + 1.01·8-s + 0.333·9-s − 0.759·11-s + 0.939·12-s − 1.23·13-s − 2.74·14-s + 0.0195·16-s − 1.30·17-s + 0.540·18-s + 0.944·19-s − 0.976·21-s − 1.23·22-s − 0.450·23-s + 0.586·24-s − 1.99·26-s + 0.192·27-s − 2.75·28-s + 0.140·29-s + 0.493·31-s − 0.984·32-s − 0.438·33-s − 2.10·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 47 | \( 1 - T \) |
| good | 2 | \( 1 - 2.29T + 2T^{2} \) |
| 7 | \( 1 + 4.47T + 7T^{2} \) |
| 11 | \( 1 + 2.51T + 11T^{2} \) |
| 13 | \( 1 + 4.43T + 13T^{2} \) |
| 17 | \( 1 + 5.36T + 17T^{2} \) |
| 19 | \( 1 - 4.11T + 19T^{2} \) |
| 23 | \( 1 + 2.16T + 23T^{2} \) |
| 29 | \( 1 - 0.758T + 29T^{2} \) |
| 31 | \( 1 - 2.74T + 31T^{2} \) |
| 37 | \( 1 - 3.99T + 37T^{2} \) |
| 41 | \( 1 + 2.29T + 41T^{2} \) |
| 43 | \( 1 + 11.8T + 43T^{2} \) |
| 53 | \( 1 + 4.40T + 53T^{2} \) |
| 59 | \( 1 + 6.47T + 59T^{2} \) |
| 61 | \( 1 - 15.3T + 61T^{2} \) |
| 67 | \( 1 + 1.66T + 67T^{2} \) |
| 71 | \( 1 - 9.28T + 71T^{2} \) |
| 73 | \( 1 + 0.677T + 73T^{2} \) |
| 79 | \( 1 + 11.5T + 79T^{2} \) |
| 83 | \( 1 + 14.1T + 83T^{2} \) |
| 89 | \( 1 - 7.01T + 89T^{2} \) |
| 97 | \( 1 - 5.26T + 97T^{2} \) |
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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
−8.005741275435865680339615555845, −7.02112481277319473488282359331, −6.71120827592363456332062095336, −5.83789228821795817871233990258, −5.02963786433072044884313505895, −4.32210121874161897486234177423, −3.40410185628943160751284430802, −2.84377234752044717875264665695, −2.21416178455403333765098304033, 0,
2.21416178455403333765098304033, 2.84377234752044717875264665695, 3.40410185628943160751284430802, 4.32210121874161897486234177423, 5.02963786433072044884313505895, 5.83789228821795817871233990258, 6.71120827592363456332062095336, 7.02112481277319473488282359331, 8.005741275435865680339615555845
