| L(s) = 1 | + 2.02·2-s + 3-s + 2.11·4-s − 5-s + 2.02·6-s − 3.74·7-s + 0.225·8-s + 9-s − 2.02·10-s + 5.54·11-s + 2.11·12-s + 3.99·13-s − 7.59·14-s − 15-s − 3.76·16-s − 5.08·17-s + 2.02·18-s − 1.91·19-s − 2.11·20-s − 3.74·21-s + 11.2·22-s + 0.225·24-s + 25-s + 8.10·26-s + 27-s − 7.91·28-s + 5.51·29-s + ⋯ |
| L(s) = 1 | + 1.43·2-s + 0.577·3-s + 1.05·4-s − 0.447·5-s + 0.827·6-s − 1.41·7-s + 0.0797·8-s + 0.333·9-s − 0.641·10-s + 1.67·11-s + 0.609·12-s + 1.10·13-s − 2.03·14-s − 0.258·15-s − 0.941·16-s − 1.23·17-s + 0.477·18-s − 0.440·19-s − 0.472·20-s − 0.817·21-s + 2.39·22-s + 0.0460·24-s + 0.200·25-s + 1.59·26-s + 0.192·27-s − 1.49·28-s + 1.02·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.702036073\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.702036073\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 - 2.02T + 2T^{2} \) |
| 7 | \( 1 + 3.74T + 7T^{2} \) |
| 11 | \( 1 - 5.54T + 11T^{2} \) |
| 13 | \( 1 - 3.99T + 13T^{2} \) |
| 17 | \( 1 + 5.08T + 17T^{2} \) |
| 19 | \( 1 + 1.91T + 19T^{2} \) |
| 29 | \( 1 - 5.51T + 29T^{2} \) |
| 31 | \( 1 - 0.180T + 31T^{2} \) |
| 37 | \( 1 - 4.47T + 37T^{2} \) |
| 41 | \( 1 - 11.6T + 41T^{2} \) |
| 43 | \( 1 - 5.14T + 43T^{2} \) |
| 47 | \( 1 - 3.81T + 47T^{2} \) |
| 53 | \( 1 - 2.53T + 53T^{2} \) |
| 59 | \( 1 + 14.2T + 59T^{2} \) |
| 61 | \( 1 + 6.52T + 61T^{2} \) |
| 67 | \( 1 - 6.37T + 67T^{2} \) |
| 71 | \( 1 - 11.9T + 71T^{2} \) |
| 73 | \( 1 - 13.7T + 73T^{2} \) |
| 79 | \( 1 - 5.17T + 79T^{2} \) |
| 83 | \( 1 - 8.33T + 83T^{2} \) |
| 89 | \( 1 + 2.12T + 89T^{2} \) |
| 97 | \( 1 - 17.8T + 97T^{2} \) |
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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
−7.66160669888680165399613700166, −6.61716996886274339631579291739, −6.47361386775026589519046470352, −5.98779194027226686396457025853, −4.70811120119687162890231789398, −4.02195682287988117126272793432, −3.78575486733487251377915392814, −3.00299763880530764563651018127, −2.24361807085514818485999058828, −0.836874475190633672161572163099,
0.836874475190633672161572163099, 2.24361807085514818485999058828, 3.00299763880530764563651018127, 3.78575486733487251377915392814, 4.02195682287988117126272793432, 4.70811120119687162890231789398, 5.98779194027226686396457025853, 6.47361386775026589519046470352, 6.61716996886274339631579291739, 7.66160669888680165399613700166
