| L(s) = 1 | + 0.687·3-s − 3.62·5-s − 7-s − 2.52·9-s + 0.456·11-s − 2.54·13-s − 2.48·15-s + 5.06·17-s + 19-s − 0.687·21-s + 3.69·23-s + 8.10·25-s − 3.80·27-s − 2.58·29-s − 6.25·31-s + 0.313·33-s + 3.62·35-s + 7.73·37-s − 1.75·39-s + 4.79·41-s + 8.17·43-s + 9.14·45-s + 12.9·47-s + 49-s + 3.48·51-s − 0.268·53-s − 1.65·55-s + ⋯ |
| L(s) = 1 | + 0.397·3-s − 1.61·5-s − 0.377·7-s − 0.842·9-s + 0.137·11-s − 0.706·13-s − 0.642·15-s + 1.22·17-s + 0.229·19-s − 0.150·21-s + 0.771·23-s + 1.62·25-s − 0.731·27-s − 0.480·29-s − 1.12·31-s + 0.0546·33-s + 0.611·35-s + 1.27·37-s − 0.280·39-s + 0.749·41-s + 1.24·43-s + 1.36·45-s + 1.88·47-s + 0.142·49-s + 0.488·51-s − 0.0368·53-s − 0.222·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 0.687T + 3T^{2} \) |
| 5 | \( 1 + 3.62T + 5T^{2} \) |
| 11 | \( 1 - 0.456T + 11T^{2} \) |
| 13 | \( 1 + 2.54T + 13T^{2} \) |
| 17 | \( 1 - 5.06T + 17T^{2} \) |
| 23 | \( 1 - 3.69T + 23T^{2} \) |
| 29 | \( 1 + 2.58T + 29T^{2} \) |
| 31 | \( 1 + 6.25T + 31T^{2} \) |
| 37 | \( 1 - 7.73T + 37T^{2} \) |
| 41 | \( 1 - 4.79T + 41T^{2} \) |
| 43 | \( 1 - 8.17T + 43T^{2} \) |
| 47 | \( 1 - 12.9T + 47T^{2} \) |
| 53 | \( 1 + 0.268T + 53T^{2} \) |
| 59 | \( 1 - 1.00T + 59T^{2} \) |
| 61 | \( 1 + 3.13T + 61T^{2} \) |
| 67 | \( 1 + 9.64T + 67T^{2} \) |
| 71 | \( 1 + 9.65T + 71T^{2} \) |
| 73 | \( 1 - 8.80T + 73T^{2} \) |
| 79 | \( 1 + 9.00T + 79T^{2} \) |
| 83 | \( 1 + 14.0T + 83T^{2} \) |
| 89 | \( 1 - 1.62T + 89T^{2} \) |
| 97 | \( 1 + 0.179T + 97T^{2} \) |
| 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
−7.57968155173562583490247654989, −7.11712326626395691504941366534, −5.94364557840634776730303383559, −5.42543769528608026195070715683, −4.39933993370961975628520943666, −3.82891249037810707552526049904, −3.09803482484923537543420961345, −2.56217889020549613254059383758, −1.01106124988482224862929993509, 0,
1.01106124988482224862929993509, 2.56217889020549613254059383758, 3.09803482484923537543420961345, 3.82891249037810707552526049904, 4.39933993370961975628520943666, 5.42543769528608026195070715683, 5.94364557840634776730303383559, 7.11712326626395691504941366534, 7.57968155173562583490247654989
