| L(s) = 1 | − 9.21e4·3-s − 1.86e7·5-s + 6.05e8·7-s − 1.97e9·9-s + 1.12e11·11-s + 7.55e11·13-s + 1.71e12·15-s + 1.15e13·17-s − 4.15e13·19-s − 5.57e13·21-s − 8.14e13·23-s − 1.29e14·25-s + 1.14e15·27-s + 2.46e15·29-s + 6.30e14·31-s − 1.03e16·33-s − 1.12e16·35-s + 1.40e16·37-s − 6.95e16·39-s + 1.20e17·41-s + 1.67e17·43-s + 3.67e16·45-s − 2.73e17·47-s − 1.92e17·49-s − 1.06e18·51-s + 1.24e18·53-s − 2.08e18·55-s + ⋯ |
| L(s) = 1 | − 0.900·3-s − 0.853·5-s + 0.809·7-s − 0.188·9-s + 1.30·11-s + 1.51·13-s + 0.769·15-s + 1.38·17-s − 1.55·19-s − 0.729·21-s − 0.409·23-s − 0.270·25-s + 1.07·27-s + 1.08·29-s + 0.138·31-s − 1.17·33-s − 0.691·35-s + 0.478·37-s − 1.36·39-s + 1.40·41-s + 1.18·43-s + 0.160·45-s − 0.757·47-s − 0.344·49-s − 1.24·51-s + 0.974·53-s − 1.11·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(\approx\) |
\(1.253247544\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.253247544\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + 9.21e4T + 1.04e10T^{2} \) |
| 5 | \( 1 + 1.86e7T + 4.76e14T^{2} \) |
| 7 | \( 1 - 6.05e8T + 5.58e17T^{2} \) |
| 11 | \( 1 - 1.12e11T + 7.40e21T^{2} \) |
| 13 | \( 1 - 7.55e11T + 2.47e23T^{2} \) |
| 17 | \( 1 - 1.15e13T + 6.90e25T^{2} \) |
| 19 | \( 1 + 4.15e13T + 7.14e26T^{2} \) |
| 23 | \( 1 + 8.14e13T + 3.94e28T^{2} \) |
| 29 | \( 1 - 2.46e15T + 5.13e30T^{2} \) |
| 31 | \( 1 - 6.30e14T + 2.08e31T^{2} \) |
| 37 | \( 1 - 1.40e16T + 8.55e32T^{2} \) |
| 41 | \( 1 - 1.20e17T + 7.38e33T^{2} \) |
| 43 | \( 1 - 1.67e17T + 2.00e34T^{2} \) |
| 47 | \( 1 + 2.73e17T + 1.30e35T^{2} \) |
| 53 | \( 1 - 1.24e18T + 1.62e36T^{2} \) |
| 59 | \( 1 + 3.96e17T + 1.54e37T^{2} \) |
| 61 | \( 1 - 8.53e18T + 3.10e37T^{2} \) |
| 67 | \( 1 + 1.76e19T + 2.22e38T^{2} \) |
| 71 | \( 1 - 1.03e19T + 7.52e38T^{2} \) |
| 73 | \( 1 + 1.55e19T + 1.34e39T^{2} \) |
| 79 | \( 1 - 7.30e19T + 7.08e39T^{2} \) |
| 83 | \( 1 - 8.17e19T + 1.99e40T^{2} \) |
| 89 | \( 1 + 2.23e20T + 8.65e40T^{2} \) |
| 97 | \( 1 + 2.78e20T + 5.27e41T^{2} \) |
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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
−19.38287205031715679609822463635, −17.66682061615140174157551704297, −16.32676626949742591358803633335, −14.48845091919012892171179162792, −12.02845683202389570979538543930, −11.01809197553806215554844389149, −8.329683351154990201538770306797, −6.12474486755055560722610693144, −4.07799393492705875920837587867, −0.985554205665853098874369929631,
0.985554205665853098874369929631, 4.07799393492705875920837587867, 6.12474486755055560722610693144, 8.329683351154990201538770306797, 11.01809197553806215554844389149, 12.02845683202389570979538543930, 14.48845091919012892171179162792, 16.32676626949742591358803633335, 17.66682061615140174157551704297, 19.38287205031715679609822463635
