| L(s) = 1 | + 4·2-s + 25.3·3-s + 16·4-s − 93.4·5-s + 101.·6-s − 182.·7-s + 64·8-s + 402.·9-s − 373.·10-s + 397.·11-s + 406.·12-s − 690.·13-s − 728.·14-s − 2.37e3·15-s + 256·16-s + 1.23e3·17-s + 1.60e3·18-s − 1.49e3·20-s − 4.62e3·21-s + 1.59e3·22-s + 281.·23-s + 1.62e3·24-s + 5.60e3·25-s − 2.76e3·26-s + 4.03e3·27-s − 2.91e3·28-s + 4.33e3·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.62·3-s + 0.5·4-s − 1.67·5-s + 1.15·6-s − 1.40·7-s + 0.353·8-s + 1.65·9-s − 1.18·10-s + 0.991·11-s + 0.814·12-s − 1.13·13-s − 0.993·14-s − 2.72·15-s + 0.250·16-s + 1.03·17-s + 1.16·18-s − 0.835·20-s − 2.28·21-s + 0.701·22-s + 0.111·23-s + 0.576·24-s + 1.79·25-s − 0.801·26-s + 1.06·27-s − 0.702·28-s + 0.958·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(4.186692761\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.186692761\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 25.3T + 243T^{2} \) |
| 5 | \( 1 + 93.4T + 3.12e3T^{2} \) |
| 7 | \( 1 + 182.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 397.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 690.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.23e3T + 1.41e6T^{2} \) |
| 23 | \( 1 - 281.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.33e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.68e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 3.85e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.95e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 371.T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.35e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 5.62e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.00e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.21e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 266.T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.22e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.56e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 5.65e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 8.51e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 2.16e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.11e5T + 8.58e9T^{2} \) |
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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
−9.505318454558723083916455309285, −8.744128433052460818926000719769, −7.70243790269577853822157270365, −7.30610536224631967498849582463, −6.34309024409472958623862496523, −4.64201304268468698527083043841, −3.78854107368533472463290452419, −3.30054700255248382263211941699, −2.53241832300229185425551750697, −0.78176344917230869575948554531,
0.78176344917230869575948554531, 2.53241832300229185425551750697, 3.30054700255248382263211941699, 3.78854107368533472463290452419, 4.64201304268468698527083043841, 6.34309024409472958623862496523, 7.30610536224631967498849582463, 7.70243790269577853822157270365, 8.744128433052460818926000719769, 9.505318454558723083916455309285
