| L(s) = 1 | − 2.23·2-s + 1.36·3-s + 2.98·4-s − 4.28·5-s − 3.03·6-s − 0.384·7-s − 2.19·8-s − 1.14·9-s + 9.56·10-s − 11-s + 4.06·12-s + 0.857·14-s − 5.83·15-s − 1.07·16-s − 4.75·17-s + 2.55·18-s + 2.17·19-s − 12.7·20-s − 0.523·21-s + 2.23·22-s + 2.70·23-s − 2.98·24-s + 13.3·25-s − 5.64·27-s − 1.14·28-s − 10.2·29-s + 13.0·30-s + ⋯ |
| L(s) = 1 | − 1.57·2-s + 0.786·3-s + 1.49·4-s − 1.91·5-s − 1.24·6-s − 0.145·7-s − 0.774·8-s − 0.381·9-s + 3.02·10-s − 0.301·11-s + 1.17·12-s + 0.229·14-s − 1.50·15-s − 0.268·16-s − 1.15·17-s + 0.602·18-s + 0.498·19-s − 2.85·20-s − 0.114·21-s + 0.475·22-s + 0.563·23-s − 0.609·24-s + 2.67·25-s − 1.08·27-s − 0.216·28-s − 1.90·29-s + 2.37·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3416881089\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3416881089\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 2.23T + 2T^{2} \) |
| 3 | \( 1 - 1.36T + 3T^{2} \) |
| 5 | \( 1 + 4.28T + 5T^{2} \) |
| 7 | \( 1 + 0.384T + 7T^{2} \) |
| 17 | \( 1 + 4.75T + 17T^{2} \) |
| 19 | \( 1 - 2.17T + 19T^{2} \) |
| 23 | \( 1 - 2.70T + 23T^{2} \) |
| 29 | \( 1 + 10.2T + 29T^{2} \) |
| 31 | \( 1 + 3.64T + 31T^{2} \) |
| 37 | \( 1 - 4.30T + 37T^{2} \) |
| 41 | \( 1 + 6.28T + 41T^{2} \) |
| 43 | \( 1 - 0.639T + 43T^{2} \) |
| 47 | \( 1 + 4.82T + 47T^{2} \) |
| 53 | \( 1 - 3.32T + 53T^{2} \) |
| 59 | \( 1 - 8.90T + 59T^{2} \) |
| 61 | \( 1 - 8.06T + 61T^{2} \) |
| 67 | \( 1 + 14.7T + 67T^{2} \) |
| 71 | \( 1 - 13.9T + 71T^{2} \) |
| 73 | \( 1 - 11.4T + 73T^{2} \) |
| 79 | \( 1 - 7.58T + 79T^{2} \) |
| 83 | \( 1 - 13.8T + 83T^{2} \) |
| 89 | \( 1 + 4.74T + 89T^{2} \) |
| 97 | \( 1 + 15.3T + 97T^{2} \) |
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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
−9.026367031844609717095844224514, −8.453326118520410508507028593028, −7.88729366531417824027125960880, −7.36417973419014732951722814691, −6.66240740135606158999561520190, −5.09543976755053292976534569178, −3.93923218662442857261951545270, −3.19199515997804463334106317491, −2.09383719015236136764397040128, −0.45824369495247081614916489734,
0.45824369495247081614916489734, 2.09383719015236136764397040128, 3.19199515997804463334106317491, 3.93923218662442857261951545270, 5.09543976755053292976534569178, 6.66240740135606158999561520190, 7.36417973419014732951722814691, 7.88729366531417824027125960880, 8.453326118520410508507028593028, 9.026367031844609717095844224514
