| L(s) = 1 | − 1.37·2-s + 3-s − 0.119·4-s + 5-s − 1.37·6-s − 3.47·7-s + 2.90·8-s + 9-s − 1.37·10-s + 2.80·11-s − 0.119·12-s + 3.04·13-s + 4.76·14-s + 15-s − 3.74·16-s + 5.76·17-s − 1.37·18-s − 0.670·19-s − 0.119·20-s − 3.47·21-s − 3.84·22-s + 2.90·24-s + 25-s − 4.17·26-s + 27-s + 0.415·28-s − 5.26·29-s + ⋯ |
| L(s) = 1 | − 0.969·2-s + 0.577·3-s − 0.0598·4-s + 0.447·5-s − 0.559·6-s − 1.31·7-s + 1.02·8-s + 0.333·9-s − 0.433·10-s + 0.846·11-s − 0.0345·12-s + 0.843·13-s + 1.27·14-s + 0.258·15-s − 0.936·16-s + 1.39·17-s − 0.323·18-s − 0.153·19-s − 0.0267·20-s − 0.757·21-s − 0.820·22-s + 0.593·24-s + 0.200·25-s − 0.818·26-s + 0.192·27-s + 0.0785·28-s − 0.977·29-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
| \(L(1)\) |
\(\approx\) |
\(1.528642190\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.528642190\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.37T + 2T^{2} \) |
| 7 | \( 1 + 3.47T + 7T^{2} \) |
| 11 | \( 1 - 2.80T + 11T^{2} \) |
| 13 | \( 1 - 3.04T + 13T^{2} \) |
| 17 | \( 1 - 5.76T + 17T^{2} \) |
| 19 | \( 1 + 0.670T + 19T^{2} \) |
| 29 | \( 1 + 5.26T + 29T^{2} \) |
| 31 | \( 1 - 7.53T + 31T^{2} \) |
| 37 | \( 1 - 6.45T + 37T^{2} \) |
| 41 | \( 1 - 2.07T + 41T^{2} \) |
| 43 | \( 1 + 2.75T + 43T^{2} \) |
| 47 | \( 1 - 5.29T + 47T^{2} \) |
| 53 | \( 1 - 0.442T + 53T^{2} \) |
| 59 | \( 1 + 14.0T + 59T^{2} \) |
| 61 | \( 1 - 11.1T + 61T^{2} \) |
| 67 | \( 1 - 1.42T + 67T^{2} \) |
| 71 | \( 1 + 13.1T + 71T^{2} \) |
| 73 | \( 1 - 8.08T + 73T^{2} \) |
| 79 | \( 1 - 5.53T + 79T^{2} \) |
| 83 | \( 1 - 2.35T + 83T^{2} \) |
| 89 | \( 1 - 15.0T + 89T^{2} \) |
| 97 | \( 1 - 10.0T + 97T^{2} \) |
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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
−7.88312938240603464148089023573, −7.42715892730346919857169489620, −6.42114882277470761564931731121, −6.12564229324876384930781854387, −5.05254983691572715496634726832, −4.00762534769295276990299800323, −3.51870962874936302668469602641, −2.61155798734546493764150112408, −1.48844691913642887206227190578, −0.75964880610399439581095003410,
0.75964880610399439581095003410, 1.48844691913642887206227190578, 2.61155798734546493764150112408, 3.51870962874936302668469602641, 4.00762534769295276990299800323, 5.05254983691572715496634726832, 6.12564229324876384930781854387, 6.42114882277470761564931731121, 7.42715892730346919857169489620, 7.88312938240603464148089023573
