| L(s) = 1 | − 0.494·2-s − 3-s − 1.75·4-s − 1.64·5-s + 0.494·6-s + 7-s + 1.85·8-s + 9-s + 0.813·10-s − 1.00·11-s + 1.75·12-s − 0.494·14-s + 1.64·15-s + 2.59·16-s − 4.38·17-s − 0.494·18-s − 1.17·19-s + 2.89·20-s − 21-s + 0.497·22-s + 2.62·23-s − 1.85·24-s − 2.28·25-s − 27-s − 1.75·28-s + 5.13·29-s − 0.813·30-s + ⋯ |
| L(s) = 1 | − 0.349·2-s − 0.577·3-s − 0.877·4-s − 0.736·5-s + 0.201·6-s + 0.377·7-s + 0.656·8-s + 0.333·9-s + 0.257·10-s − 0.303·11-s + 0.506·12-s − 0.132·14-s + 0.425·15-s + 0.648·16-s − 1.06·17-s − 0.116·18-s − 0.269·19-s + 0.646·20-s − 0.218·21-s + 0.106·22-s + 0.548·23-s − 0.378·24-s − 0.457·25-s − 0.192·27-s − 0.331·28-s + 0.954·29-s − 0.148·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5312655157\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5312655157\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 0.494T + 2T^{2} \) |
| 5 | \( 1 + 1.64T + 5T^{2} \) |
| 11 | \( 1 + 1.00T + 11T^{2} \) |
| 17 | \( 1 + 4.38T + 17T^{2} \) |
| 19 | \( 1 + 1.17T + 19T^{2} \) |
| 23 | \( 1 - 2.62T + 23T^{2} \) |
| 29 | \( 1 - 5.13T + 29T^{2} \) |
| 31 | \( 1 - 0.705T + 31T^{2} \) |
| 37 | \( 1 + 2.72T + 37T^{2} \) |
| 41 | \( 1 - 3.05T + 41T^{2} \) |
| 43 | \( 1 + 10.5T + 43T^{2} \) |
| 47 | \( 1 - 2.16T + 47T^{2} \) |
| 53 | \( 1 + 1.96T + 53T^{2} \) |
| 59 | \( 1 - 4.90T + 59T^{2} \) |
| 61 | \( 1 - 0.563T + 61T^{2} \) |
| 67 | \( 1 + 13.1T + 67T^{2} \) |
| 71 | \( 1 + 15.3T + 71T^{2} \) |
| 73 | \( 1 + 13.5T + 73T^{2} \) |
| 79 | \( 1 - 7.63T + 79T^{2} \) |
| 83 | \( 1 + 0.0542T + 83T^{2} \) |
| 89 | \( 1 - 0.601T + 89T^{2} \) |
| 97 | \( 1 + 1.74T + 97T^{2} \) |
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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
−8.635867014337867201644585867064, −7.87941761220679732252172846359, −7.25194598380720463727351627625, −6.36667670073228606159190784097, −5.41240358486337247215365198975, −4.61549975177559331478539223747, −4.23616110537515169624515375359, −3.13176852576993708208664562376, −1.72078092434549326153685728126, −0.47391784647189520176447917843,
0.47391784647189520176447917843, 1.72078092434549326153685728126, 3.13176852576993708208664562376, 4.23616110537515169624515375359, 4.61549975177559331478539223747, 5.41240358486337247215365198975, 6.36667670073228606159190784097, 7.25194598380720463727351627625, 7.87941761220679732252172846359, 8.635867014337867201644585867064
