| L(s) = 1 | − 1.79·3-s + 4.30·5-s + 0.947·7-s + 0.217·9-s + 3.02·11-s + 3.61·13-s − 7.71·15-s − 3.28·17-s − 19-s − 1.70·21-s − 4.49·23-s + 13.4·25-s + 4.99·27-s − 6.42·29-s − 0.341·31-s − 5.42·33-s + 4.07·35-s + 2.28·37-s − 6.47·39-s + 10.1·41-s + 4.04·43-s + 0.935·45-s + 0.584·47-s − 6.10·49-s + 5.88·51-s − 0.0341·53-s + 13.0·55-s + ⋯ |
| L(s) = 1 | − 1.03·3-s + 1.92·5-s + 0.358·7-s + 0.0725·9-s + 0.911·11-s + 1.00·13-s − 1.99·15-s − 0.796·17-s − 0.229·19-s − 0.371·21-s − 0.936·23-s + 2.69·25-s + 0.960·27-s − 1.19·29-s − 0.0613·31-s − 0.944·33-s + 0.689·35-s + 0.375·37-s − 1.03·39-s + 1.58·41-s + 0.616·43-s + 0.139·45-s + 0.0852·47-s − 0.871·49-s + 0.824·51-s − 0.00469·53-s + 1.75·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.293864683\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.293864683\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + T \) |
| 79 | \( 1 + T \) |
| good | 3 | \( 1 + 1.79T + 3T^{2} \) |
| 5 | \( 1 - 4.30T + 5T^{2} \) |
| 7 | \( 1 - 0.947T + 7T^{2} \) |
| 11 | \( 1 - 3.02T + 11T^{2} \) |
| 13 | \( 1 - 3.61T + 13T^{2} \) |
| 17 | \( 1 + 3.28T + 17T^{2} \) |
| 23 | \( 1 + 4.49T + 23T^{2} \) |
| 29 | \( 1 + 6.42T + 29T^{2} \) |
| 31 | \( 1 + 0.341T + 31T^{2} \) |
| 37 | \( 1 - 2.28T + 37T^{2} \) |
| 41 | \( 1 - 10.1T + 41T^{2} \) |
| 43 | \( 1 - 4.04T + 43T^{2} \) |
| 47 | \( 1 - 0.584T + 47T^{2} \) |
| 53 | \( 1 + 0.0341T + 53T^{2} \) |
| 59 | \( 1 - 12.0T + 59T^{2} \) |
| 61 | \( 1 - 9.19T + 61T^{2} \) |
| 67 | \( 1 + 15.4T + 67T^{2} \) |
| 71 | \( 1 - 0.662T + 71T^{2} \) |
| 73 | \( 1 - 11.2T + 73T^{2} \) |
| 83 | \( 1 - 4.85T + 83T^{2} \) |
| 89 | \( 1 - 15.3T + 89T^{2} \) |
| 97 | \( 1 + 10.9T + 97T^{2} \) |
| 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
−8.202073201233260131742794275856, −7.02404294062916999095369977121, −6.31906077787782489675331508197, −6.00674607840994760373585644372, −5.48987338723304484282911420372, −4.65915052583132190090386729858, −3.79167571626337233154249789110, −2.47406534791725134219739423038, −1.76585013796199854617228597680, −0.883962849605260431862945591255,
0.883962849605260431862945591255, 1.76585013796199854617228597680, 2.47406534791725134219739423038, 3.79167571626337233154249789110, 4.65915052583132190090386729858, 5.48987338723304484282911420372, 6.00674607840994760373585644372, 6.31906077787782489675331508197, 7.02404294062916999095369977121, 8.202073201233260131742794275856
