| L(s) = 1 | + 10.9·2-s + 11.8·3-s + 87.9·4-s + 129.·6-s − 32.4·7-s + 613.·8-s − 103.·9-s + 452.·11-s + 1.03e3·12-s − 141.·13-s − 355.·14-s + 3.89e3·16-s + 675.·17-s − 1.13e3·18-s + 2.39e3·19-s − 383.·21-s + 4.95e3·22-s − 555.·23-s + 7.24e3·24-s − 1.55e3·26-s − 4.09e3·27-s − 2.85e3·28-s + 3.18e3·29-s − 5.90e3·31-s + 2.30e4·32-s + 5.34e3·33-s + 7.39e3·34-s + ⋯ |
| L(s) = 1 | + 1.93·2-s + 0.757·3-s + 2.74·4-s + 1.46·6-s − 0.250·7-s + 3.38·8-s − 0.425·9-s + 1.12·11-s + 2.08·12-s − 0.232·13-s − 0.484·14-s + 3.80·16-s + 0.566·17-s − 0.824·18-s + 1.51·19-s − 0.189·21-s + 2.18·22-s − 0.219·23-s + 2.56·24-s − 0.450·26-s − 1.08·27-s − 0.688·28-s + 0.704·29-s − 1.10·31-s + 3.98·32-s + 0.854·33-s + 1.09·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(13.10035240\) |
| \(L(\frac12)\) |
\(\approx\) |
\(13.10035240\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 43 | \( 1 + 1.84e3T \) |
| good | 2 | \( 1 - 10.9T + 32T^{2} \) |
| 3 | \( 1 - 11.8T + 243T^{2} \) |
| 7 | \( 1 + 32.4T + 1.68e4T^{2} \) |
| 11 | \( 1 - 452.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 141.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 675.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.39e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 555.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 3.18e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 5.90e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 6.79e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.47e4T + 1.15e8T^{2} \) |
| 47 | \( 1 + 1.75e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 6.27e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.51e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.20e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.83e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.72e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.05e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.34e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 8.91e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 9.80e3T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.52e5T + 8.58e9T^{2} \) |
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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
−9.239637056484848080642927044140, −7.971003017441855993576778686962, −7.29079484791851564576295891833, −6.35573136385516906299772187640, −5.65414384849925804951441685055, −4.75586335843334603538127892401, −3.68935055295407225902954523360, −3.24101461768821720741135462564, −2.32670685604634833290988294102, −1.22276585657651607735038720015,
1.22276585657651607735038720015, 2.32670685604634833290988294102, 3.24101461768821720741135462564, 3.68935055295407225902954523360, 4.75586335843334603538127892401, 5.65414384849925804951441685055, 6.35573136385516906299772187640, 7.29079484791851564576295891833, 7.971003017441855993576778686962, 9.239637056484848080642927044140
