| L(s) = 1 | − 8.48·5-s + 7·7-s − 25.4·13-s − 8.48·19-s − 10·23-s + 46.9·25-s − 59.3·35-s + 49·49-s + 76.3·59-s + 8.48·61-s + 215.·65-s − 110·71-s − 130·79-s − 25.4·83-s − 178.·91-s + 71.9·95-s + 161.·101-s + 26·113-s + 84.8·115-s + ⋯ |
| L(s) = 1 | − 1.69·5-s + 7-s − 1.95·13-s − 0.446·19-s − 0.434·23-s + 1.87·25-s − 1.69·35-s + 0.999·49-s + 1.29·59-s + 0.139·61-s + 3.32·65-s − 1.54·71-s − 1.64·79-s − 0.306·83-s − 1.95·91-s + 0.757·95-s + 1.59·101-s + 0.230·113-s + 0.737·115-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9626579240\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9626579240\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| good | 5 | \( 1 + 8.48T + 25T^{2} \) |
| 11 | \( 1 - 121T^{2} \) |
| 13 | \( 1 + 25.4T + 169T^{2} \) |
| 17 | \( 1 - 289T^{2} \) |
| 19 | \( 1 + 8.48T + 361T^{2} \) |
| 23 | \( 1 + 10T + 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 - 1.36e3T^{2} \) |
| 41 | \( 1 - 1.68e3T^{2} \) |
| 43 | \( 1 - 1.84e3T^{2} \) |
| 47 | \( 1 - 2.20e3T^{2} \) |
| 53 | \( 1 - 2.80e3T^{2} \) |
| 59 | \( 1 - 76.3T + 3.48e3T^{2} \) |
| 61 | \( 1 - 8.48T + 3.72e3T^{2} \) |
| 67 | \( 1 - 4.48e3T^{2} \) |
| 71 | \( 1 + 110T + 5.04e3T^{2} \) |
| 73 | \( 1 - 5.32e3T^{2} \) |
| 79 | \( 1 + 130T + 6.24e3T^{2} \) |
| 83 | \( 1 + 25.4T + 6.88e3T^{2} \) |
| 89 | \( 1 - 7.92e3T^{2} \) |
| 97 | \( 1 - 9.40e3T^{2} \) |
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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
−8.679437747347805917702158442653, −8.142162648135797549346924437916, −7.39744139844505844982967995423, −7.05254454948978585113206874928, −5.60421667357358108436214468828, −4.61720476533318288255584822129, −4.30760384176978428993386361777, −3.13110392646624773242943929686, −2.05597076611335578342927298225, −0.50606179316495413297617823634,
0.50606179316495413297617823634, 2.05597076611335578342927298225, 3.13110392646624773242943929686, 4.30760384176978428993386361777, 4.61720476533318288255584822129, 5.60421667357358108436214468828, 7.05254454948978585113206874928, 7.39744139844505844982967995423, 8.142162648135797549346924437916, 8.679437747347805917702158442653
