| L(s) = 1 | + 2.08·3-s + 5-s − 3.25·7-s + 1.36·9-s + 5.61·11-s + 2.90·13-s + 2.08·15-s + 5.25·17-s + 4.99·19-s − 6.79·21-s + 7.72·23-s + 25-s − 3.41·27-s − 29-s − 4.73·31-s + 11.7·33-s − 3.25·35-s + 6.62·37-s + 6.06·39-s − 4.43·41-s − 4.30·43-s + 1.36·45-s − 2.19·47-s + 3.57·49-s + 10.9·51-s − 5.81·53-s + 5.61·55-s + ⋯ |
| L(s) = 1 | + 1.20·3-s + 0.447·5-s − 1.22·7-s + 0.454·9-s + 1.69·11-s + 0.805·13-s + 0.539·15-s + 1.27·17-s + 1.14·19-s − 1.48·21-s + 1.60·23-s + 0.200·25-s − 0.657·27-s − 0.185·29-s − 0.850·31-s + 2.04·33-s − 0.549·35-s + 1.08·37-s + 0.971·39-s − 0.692·41-s − 0.656·43-s + 0.203·45-s − 0.319·47-s + 0.511·49-s + 1.53·51-s − 0.798·53-s + 0.757·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.062600846\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.062600846\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 - 2.08T + 3T^{2} \) |
| 7 | \( 1 + 3.25T + 7T^{2} \) |
| 11 | \( 1 - 5.61T + 11T^{2} \) |
| 13 | \( 1 - 2.90T + 13T^{2} \) |
| 17 | \( 1 - 5.25T + 17T^{2} \) |
| 19 | \( 1 - 4.99T + 19T^{2} \) |
| 23 | \( 1 - 7.72T + 23T^{2} \) |
| 31 | \( 1 + 4.73T + 31T^{2} \) |
| 37 | \( 1 - 6.62T + 37T^{2} \) |
| 41 | \( 1 + 4.43T + 41T^{2} \) |
| 43 | \( 1 + 4.30T + 43T^{2} \) |
| 47 | \( 1 + 2.19T + 47T^{2} \) |
| 53 | \( 1 + 5.81T + 53T^{2} \) |
| 59 | \( 1 - 8.90T + 59T^{2} \) |
| 61 | \( 1 + 13.2T + 61T^{2} \) |
| 67 | \( 1 + 12.1T + 67T^{2} \) |
| 71 | \( 1 + 1.31T + 71T^{2} \) |
| 73 | \( 1 - 5.17T + 73T^{2} \) |
| 79 | \( 1 - 11.5T + 79T^{2} \) |
| 83 | \( 1 + 2.05T + 83T^{2} \) |
| 89 | \( 1 - 2.17T + 89T^{2} \) |
| 97 | \( 1 - 12.9T + 97T^{2} \) |
| 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
−7.67800962411570057105138031249, −7.09747296425966522272810614223, −6.35350241147307051092349724755, −5.87030940916234807335509434386, −4.92842583364031487868816045646, −3.75597269149834854600535351299, −3.34736579822146101331832302182, −2.96758611818389382385256044361, −1.70079701936075546483232445759, −0.990280917597518757638647771884,
0.990280917597518757638647771884, 1.70079701936075546483232445759, 2.96758611818389382385256044361, 3.34736579822146101331832302182, 3.75597269149834854600535351299, 4.92842583364031487868816045646, 5.87030940916234807335509434386, 6.35350241147307051092349724755, 7.09747296425966522272810614223, 7.67800962411570057105138031249
