| L(s) = 1 | − 0.618·3-s + 1.23·5-s − 4.23·7-s − 2.61·9-s + 6.61·11-s − 5.47·13-s − 0.763·15-s + 2.76·17-s + 2.61·21-s + 0.854·23-s − 3.47·25-s + 3.47·27-s + 3.85·29-s + 6.61·31-s − 4.09·33-s − 5.23·35-s − 8.61·37-s + 3.38·39-s − 9.94·41-s − 1.61·43-s − 3.23·45-s + 0.708·47-s + 10.9·49-s − 1.70·51-s − 2.85·53-s + 8.18·55-s + 7.61·59-s + ⋯ |
| L(s) = 1 | − 0.356·3-s + 0.552·5-s − 1.60·7-s − 0.872·9-s + 1.99·11-s − 1.51·13-s − 0.197·15-s + 0.670·17-s + 0.571·21-s + 0.178·23-s − 0.694·25-s + 0.668·27-s + 0.715·29-s + 1.18·31-s − 0.712·33-s − 0.885·35-s − 1.41·37-s + 0.541·39-s − 1.55·41-s − 0.246·43-s − 0.482·45-s + 0.103·47-s + 1.56·49-s − 0.239·51-s − 0.392·53-s + 1.10·55-s + 0.991·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.183777722\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.183777722\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 0.618T + 3T^{2} \) |
| 5 | \( 1 - 1.23T + 5T^{2} \) |
| 7 | \( 1 + 4.23T + 7T^{2} \) |
| 11 | \( 1 - 6.61T + 11T^{2} \) |
| 13 | \( 1 + 5.47T + 13T^{2} \) |
| 17 | \( 1 - 2.76T + 17T^{2} \) |
| 23 | \( 1 - 0.854T + 23T^{2} \) |
| 29 | \( 1 - 3.85T + 29T^{2} \) |
| 31 | \( 1 - 6.61T + 31T^{2} \) |
| 37 | \( 1 + 8.61T + 37T^{2} \) |
| 41 | \( 1 + 9.94T + 41T^{2} \) |
| 43 | \( 1 + 1.61T + 43T^{2} \) |
| 47 | \( 1 - 0.708T + 47T^{2} \) |
| 53 | \( 1 + 2.85T + 53T^{2} \) |
| 59 | \( 1 - 7.61T + 59T^{2} \) |
| 61 | \( 1 + 6.70T + 61T^{2} \) |
| 67 | \( 1 + 6.70T + 67T^{2} \) |
| 71 | \( 1 + 7.18T + 71T^{2} \) |
| 73 | \( 1 - 12.2T + 73T^{2} \) |
| 79 | \( 1 - 6T + 79T^{2} \) |
| 83 | \( 1 + 2.94T + 83T^{2} \) |
| 89 | \( 1 - 6.70T + 89T^{2} \) |
| 97 | \( 1 + 0.673T + 97T^{2} \) |
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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
−8.216982973609517951178856670330, −7.02419819730126784505927377563, −6.66172494461054864273599653624, −6.09505023912583298414515649869, −5.39754422104235225030781536727, −4.50953974210629024807846354879, −3.46181705524464182698560446310, −2.97244079927548007476813659280, −1.86358858183971188358940607968, −0.57118606803571554812232021716,
0.57118606803571554812232021716, 1.86358858183971188358940607968, 2.97244079927548007476813659280, 3.46181705524464182698560446310, 4.50953974210629024807846354879, 5.39754422104235225030781536727, 6.09505023912583298414515649869, 6.66172494461054864273599653624, 7.02419819730126784505927377563, 8.216982973609517951178856670330
