| L(s) = 1 | + 2.82·3-s − 2.81·5-s + 4.46·7-s + 4.99·9-s − 2.83·11-s − 3.16·13-s − 7.96·15-s − 17-s + 6.24·19-s + 12.6·21-s + 3.47·23-s + 2.92·25-s + 5.65·27-s + 4.26·29-s + 2.55·31-s − 8.01·33-s − 12.5·35-s − 5.85·37-s − 8.96·39-s − 10.2·41-s + 7.92·43-s − 14.0·45-s + 2.40·47-s + 12.9·49-s − 2.82·51-s + 11.7·53-s + 7.97·55-s + ⋯ |
| L(s) = 1 | + 1.63·3-s − 1.25·5-s + 1.68·7-s + 1.66·9-s − 0.854·11-s − 0.879·13-s − 2.05·15-s − 0.242·17-s + 1.43·19-s + 2.75·21-s + 0.724·23-s + 0.584·25-s + 1.08·27-s + 0.791·29-s + 0.459·31-s − 1.39·33-s − 2.12·35-s − 0.961·37-s − 1.43·39-s − 1.60·41-s + 1.20·43-s − 2.09·45-s + 0.350·47-s + 1.85·49-s − 0.396·51-s + 1.61·53-s + 1.07·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.665895552\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.665895552\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 - T \) |
| good | 3 | \( 1 - 2.82T + 3T^{2} \) |
| 5 | \( 1 + 2.81T + 5T^{2} \) |
| 7 | \( 1 - 4.46T + 7T^{2} \) |
| 11 | \( 1 + 2.83T + 11T^{2} \) |
| 13 | \( 1 + 3.16T + 13T^{2} \) |
| 19 | \( 1 - 6.24T + 19T^{2} \) |
| 23 | \( 1 - 3.47T + 23T^{2} \) |
| 29 | \( 1 - 4.26T + 29T^{2} \) |
| 31 | \( 1 - 2.55T + 31T^{2} \) |
| 37 | \( 1 + 5.85T + 37T^{2} \) |
| 41 | \( 1 + 10.2T + 41T^{2} \) |
| 43 | \( 1 - 7.92T + 43T^{2} \) |
| 47 | \( 1 - 2.40T + 47T^{2} \) |
| 53 | \( 1 - 11.7T + 53T^{2} \) |
| 61 | \( 1 - 10.9T + 61T^{2} \) |
| 67 | \( 1 - 8.75T + 67T^{2} \) |
| 71 | \( 1 + 16.3T + 71T^{2} \) |
| 73 | \( 1 + 0.354T + 73T^{2} \) |
| 79 | \( 1 - 6.23T + 79T^{2} \) |
| 83 | \( 1 - 12.3T + 83T^{2} \) |
| 89 | \( 1 + 9.12T + 89T^{2} \) |
| 97 | \( 1 + 5.00T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.87489880211471929213299187110, −7.37000798896189433281628922068, −7.09629340387294374620523935033, −5.36239674246265235427064148445, −4.88115483854681197338512679839, −4.20636020635798012270349703924, −3.43287785818925233513954536374, −2.70618437102495430513814963530, −2.00960872768679769510749499664, −0.894270092114901578250997579622,
0.894270092114901578250997579622, 2.00960872768679769510749499664, 2.70618437102495430513814963530, 3.43287785818925233513954536374, 4.20636020635798012270349703924, 4.88115483854681197338512679839, 5.36239674246265235427064148445, 7.09629340387294374620523935033, 7.37000798896189433281628922068, 7.87489880211471929213299187110
