| L(s) = 1 | + 1.31·3-s − 5-s + 4.66·7-s − 1.28·9-s − 2.23·11-s − 2.80·13-s − 1.31·15-s + 7.63·17-s + 1.36·19-s + 6.11·21-s + 23-s + 25-s − 5.61·27-s + 8.94·29-s + 1.58·31-s − 2.92·33-s − 4.66·35-s − 1.40·37-s − 3.67·39-s + 10.7·41-s + 1.28·45-s + 7.26·47-s + 14.7·49-s + 10.0·51-s − 8.38·53-s + 2.23·55-s + 1.78·57-s + ⋯ |
| L(s) = 1 | + 0.756·3-s − 0.447·5-s + 1.76·7-s − 0.427·9-s − 0.672·11-s − 0.776·13-s − 0.338·15-s + 1.85·17-s + 0.312·19-s + 1.33·21-s + 0.208·23-s + 0.200·25-s − 1.08·27-s + 1.66·29-s + 0.284·31-s − 0.508·33-s − 0.788·35-s − 0.230·37-s − 0.587·39-s + 1.67·41-s + 0.191·45-s + 1.05·47-s + 2.10·49-s + 1.40·51-s − 1.15·53-s + 0.300·55-s + 0.236·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.450853463\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.450853463\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 - 1.31T + 3T^{2} \) |
| 7 | \( 1 - 4.66T + 7T^{2} \) |
| 11 | \( 1 + 2.23T + 11T^{2} \) |
| 13 | \( 1 + 2.80T + 13T^{2} \) |
| 17 | \( 1 - 7.63T + 17T^{2} \) |
| 19 | \( 1 - 1.36T + 19T^{2} \) |
| 29 | \( 1 - 8.94T + 29T^{2} \) |
| 31 | \( 1 - 1.58T + 31T^{2} \) |
| 37 | \( 1 + 1.40T + 37T^{2} \) |
| 41 | \( 1 - 10.7T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 7.26T + 47T^{2} \) |
| 53 | \( 1 + 8.38T + 53T^{2} \) |
| 59 | \( 1 - 4.88T + 59T^{2} \) |
| 61 | \( 1 - 4.33T + 61T^{2} \) |
| 67 | \( 1 + 8.54T + 67T^{2} \) |
| 71 | \( 1 + 8.81T + 71T^{2} \) |
| 73 | \( 1 + 5.26T + 73T^{2} \) |
| 79 | \( 1 + 7.08T + 79T^{2} \) |
| 83 | \( 1 - 4.59T + 83T^{2} \) |
| 89 | \( 1 + 4.70T + 89T^{2} \) |
| 97 | \( 1 - 16.5T + 97T^{2} \) |
| 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
−9.016582168823071839985359932099, −8.311455487006037228373533695297, −7.72915346197858723845644383251, −7.42892889986029071243694209985, −5.82520055560519167698584789344, −5.09825589430655363909642751251, −4.37295117066608871567554175610, −3.15386255094323939197696483215, −2.40917653730055886239035235130, −1.09845870033423592219487773786,
1.09845870033423592219487773786, 2.40917653730055886239035235130, 3.15386255094323939197696483215, 4.37295117066608871567554175610, 5.09825589430655363909642751251, 5.82520055560519167698584789344, 7.42892889986029071243694209985, 7.72915346197858723845644383251, 8.311455487006037228373533695297, 9.016582168823071839985359932099
