| L(s) = 1 | + 2.68·2-s + 3-s + 5.18·4-s − 4.39·5-s + 2.68·6-s + 4.74·7-s + 8.53·8-s + 9-s − 11.7·10-s − 2.67·11-s + 5.18·12-s − 13-s + 12.7·14-s − 4.39·15-s + 12.5·16-s + 6.95·17-s + 2.68·18-s − 5.70·19-s − 22.8·20-s + 4.74·21-s − 7.16·22-s + 3.97·23-s + 8.53·24-s + 14.3·25-s − 2.68·26-s + 27-s + 24.6·28-s + ⋯ |
| L(s) = 1 | + 1.89·2-s + 0.577·3-s + 2.59·4-s − 1.96·5-s + 1.09·6-s + 1.79·7-s + 3.01·8-s + 0.333·9-s − 3.72·10-s − 0.806·11-s + 1.49·12-s − 0.277·13-s + 3.40·14-s − 1.13·15-s + 3.12·16-s + 1.68·17-s + 0.631·18-s − 1.30·19-s − 5.09·20-s + 1.03·21-s − 1.52·22-s + 0.828·23-s + 1.74·24-s + 2.86·25-s − 0.525·26-s + 0.192·27-s + 4.65·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.298409613\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.298409613\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 + T \) |
| good | 2 | \( 1 - 2.68T + 2T^{2} \) |
| 5 | \( 1 + 4.39T + 5T^{2} \) |
| 7 | \( 1 - 4.74T + 7T^{2} \) |
| 11 | \( 1 + 2.67T + 11T^{2} \) |
| 17 | \( 1 - 6.95T + 17T^{2} \) |
| 19 | \( 1 + 5.70T + 19T^{2} \) |
| 23 | \( 1 - 3.97T + 23T^{2} \) |
| 29 | \( 1 - 1.35T + 29T^{2} \) |
| 31 | \( 1 - 0.614T + 31T^{2} \) |
| 37 | \( 1 - 6.79T + 37T^{2} \) |
| 41 | \( 1 + 7.22T + 41T^{2} \) |
| 43 | \( 1 - 3.32T + 43T^{2} \) |
| 47 | \( 1 - 9.03T + 47T^{2} \) |
| 53 | \( 1 + 4.02T + 53T^{2} \) |
| 59 | \( 1 + 12.2T + 59T^{2} \) |
| 61 | \( 1 + 7.29T + 61T^{2} \) |
| 67 | \( 1 - 6.51T + 67T^{2} \) |
| 71 | \( 1 + 10.9T + 71T^{2} \) |
| 73 | \( 1 + 0.549T + 73T^{2} \) |
| 79 | \( 1 - 1.40T + 79T^{2} \) |
| 83 | \( 1 + 4.00T + 83T^{2} \) |
| 89 | \( 1 + 15.6T + 89T^{2} \) |
| 97 | \( 1 - 14.6T + 97T^{2} \) |
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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
−8.051894474358254734322038509679, −7.54507868934809704076446138349, −7.25281418144672269018196077461, −5.98199557105860527253327383950, −4.91569006189929352322698351212, −4.70001907151080687311780573244, −3.99781797406801332882071249219, −3.22017281854010956447373907102, −2.50122649160534470260504696791, −1.26822814375353016030303546973,
1.26822814375353016030303546973, 2.50122649160534470260504696791, 3.22017281854010956447373907102, 3.99781797406801332882071249219, 4.70001907151080687311780573244, 4.91569006189929352322698351212, 5.98199557105860527253327383950, 7.25281418144672269018196077461, 7.54507868934809704076446138349, 8.051894474358254734322038509679
