| L(s) = 1 | + 0.123·2-s + 3-s − 1.98·4-s − 1.78·5-s + 0.123·6-s − 2.58·7-s − 0.491·8-s + 9-s − 0.220·10-s − 1.98·12-s + 5.17·13-s − 0.318·14-s − 1.78·15-s + 3.90·16-s − 3.40·17-s + 0.123·18-s + 19-s + 3.54·20-s − 2.58·21-s − 4.24·23-s − 0.491·24-s − 1.81·25-s + 0.637·26-s + 27-s + 5.12·28-s − 3.81·29-s − 0.220·30-s + ⋯ |
| L(s) = 1 | + 0.0872·2-s + 0.577·3-s − 0.992·4-s − 0.798·5-s + 0.0503·6-s − 0.976·7-s − 0.173·8-s + 0.333·9-s − 0.0696·10-s − 0.572·12-s + 1.43·13-s − 0.0851·14-s − 0.461·15-s + 0.977·16-s − 0.824·17-s + 0.0290·18-s + 0.229·19-s + 0.792·20-s − 0.563·21-s − 0.885·23-s − 0.100·24-s − 0.362·25-s + 0.125·26-s + 0.192·27-s + 0.968·28-s − 0.709·29-s − 0.0402·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6897 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6897 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.014381398\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.014381398\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 11 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 0.123T + 2T^{2} \) |
| 5 | \( 1 + 1.78T + 5T^{2} \) |
| 7 | \( 1 + 2.58T + 7T^{2} \) |
| 13 | \( 1 - 5.17T + 13T^{2} \) |
| 17 | \( 1 + 3.40T + 17T^{2} \) |
| 23 | \( 1 + 4.24T + 23T^{2} \) |
| 29 | \( 1 + 3.81T + 29T^{2} \) |
| 31 | \( 1 - 6.05T + 31T^{2} \) |
| 37 | \( 1 + 1.61T + 37T^{2} \) |
| 41 | \( 1 + 3.98T + 41T^{2} \) |
| 43 | \( 1 + 3.92T + 43T^{2} \) |
| 47 | \( 1 + 3.87T + 47T^{2} \) |
| 53 | \( 1 + 12.0T + 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 - 3.01T + 61T^{2} \) |
| 67 | \( 1 - 4.22T + 67T^{2} \) |
| 71 | \( 1 + 12.1T + 71T^{2} \) |
| 73 | \( 1 - 1.72T + 73T^{2} \) |
| 79 | \( 1 - 5.32T + 79T^{2} \) |
| 83 | \( 1 + 10.7T + 83T^{2} \) |
| 89 | \( 1 - 3.60T + 89T^{2} \) |
| 97 | \( 1 - 17.3T + 97T^{2} \) |
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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
−8.203207738536016504259501175634, −7.41261123547871563743847230954, −6.46510567234141230761435791234, −5.99300617150448977761360628269, −4.91728401914507986960010080787, −4.14950550435049131311425547281, −3.60120842283819384136291294290, −3.13794660727291709011822543227, −1.78243043826287142247766089679, −0.49255142401501213920579512427,
0.49255142401501213920579512427, 1.78243043826287142247766089679, 3.13794660727291709011822543227, 3.60120842283819384136291294290, 4.14950550435049131311425547281, 4.91728401914507986960010080787, 5.99300617150448977761360628269, 6.46510567234141230761435791234, 7.41261123547871563743847230954, 8.203207738536016504259501175634
