| L(s) = 1 | + 5-s + 7-s − 6.09·11-s − 2.23·13-s + 7.61·17-s − 19-s + 8.70·23-s − 4·25-s − 0.854·29-s − 2.38·31-s + 35-s − 6.70·37-s − 0.0901·41-s + 8.47·43-s − 4.70·47-s + 49-s + 14.3·53-s − 6.09·55-s + 10.7·59-s + 5.47·61-s − 2.23·65-s − 2.09·67-s + 11.1·71-s − 0.909·73-s − 6.09·77-s − 6.94·79-s − 14.6·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.377·7-s − 1.83·11-s − 0.620·13-s + 1.84·17-s − 0.229·19-s + 1.81·23-s − 0.800·25-s − 0.158·29-s − 0.427·31-s + 0.169·35-s − 1.10·37-s − 0.0140·41-s + 1.29·43-s − 0.686·47-s + 0.142·49-s + 1.96·53-s − 0.821·55-s + 1.39·59-s + 0.700·61-s − 0.277·65-s − 0.255·67-s + 1.32·71-s − 0.106·73-s − 0.694·77-s − 0.781·79-s − 1.60·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.938721016\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.938721016\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 - T + 5T^{2} \) |
| 11 | \( 1 + 6.09T + 11T^{2} \) |
| 13 | \( 1 + 2.23T + 13T^{2} \) |
| 17 | \( 1 - 7.61T + 17T^{2} \) |
| 23 | \( 1 - 8.70T + 23T^{2} \) |
| 29 | \( 1 + 0.854T + 29T^{2} \) |
| 31 | \( 1 + 2.38T + 31T^{2} \) |
| 37 | \( 1 + 6.70T + 37T^{2} \) |
| 41 | \( 1 + 0.0901T + 41T^{2} \) |
| 43 | \( 1 - 8.47T + 43T^{2} \) |
| 47 | \( 1 + 4.70T + 47T^{2} \) |
| 53 | \( 1 - 14.3T + 53T^{2} \) |
| 59 | \( 1 - 10.7T + 59T^{2} \) |
| 61 | \( 1 - 5.47T + 61T^{2} \) |
| 67 | \( 1 + 2.09T + 67T^{2} \) |
| 71 | \( 1 - 11.1T + 71T^{2} \) |
| 73 | \( 1 + 0.909T + 73T^{2} \) |
| 79 | \( 1 + 6.94T + 79T^{2} \) |
| 83 | \( 1 + 14.6T + 83T^{2} \) |
| 89 | \( 1 - 12.1T + 89T^{2} \) |
| 97 | \( 1 - 8.70T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.236199094675516919659078092017, −7.44898231417672297571748422030, −7.15686561583390945993862983723, −5.79575393009660039737125421828, −5.37421634891878140555700028969, −4.86840392221287782521119728974, −3.63009913458997148789985714402, −2.78958210214579634829878740168, −2.03847630787601743863151891662, −0.76054146354421515469178553735,
0.76054146354421515469178553735, 2.03847630787601743863151891662, 2.78958210214579634829878740168, 3.63009913458997148789985714402, 4.86840392221287782521119728974, 5.37421634891878140555700028969, 5.79575393009660039737125421828, 7.15686561583390945993862983723, 7.44898231417672297571748422030, 8.236199094675516919659078092017
