| L(s) = 1 | + 5.19·2-s − 3·3-s + 19·4-s + 9·5-s − 15.5·6-s + 24.2·7-s + 57.1·8-s + 9·9-s + 46.7·10-s − 57·12-s − 71.0·13-s + 126·14-s − 27·15-s + 145·16-s − 88.3·17-s + 46.7·18-s + 145.·19-s + 171·20-s − 72.7·21-s + 90·23-s − 171.·24-s − 44·25-s − 369·26-s − 27·27-s + 460.·28-s + 88.3·29-s − 140.·30-s + ⋯ |
| L(s) = 1 | + 1.83·2-s − 0.577·3-s + 2.37·4-s + 0.804·5-s − 1.06·6-s + 1.30·7-s + 2.52·8-s + 0.333·9-s + 1.47·10-s − 1.37·12-s − 1.51·13-s + 2.40·14-s − 0.464·15-s + 2.26·16-s − 1.26·17-s + 0.612·18-s + 1.75·19-s + 1.91·20-s − 0.755·21-s + 0.815·23-s − 1.45·24-s − 0.351·25-s − 2.78·26-s − 0.192·27-s + 3.10·28-s + 0.565·29-s − 0.853·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.083177519\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.083177519\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 5.19T + 8T^{2} \) |
| 5 | \( 1 - 9T + 125T^{2} \) |
| 7 | \( 1 - 24.2T + 343T^{2} \) |
| 13 | \( 1 + 71.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 88.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 145.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 90T + 1.21e4T^{2} \) |
| 29 | \( 1 - 88.3T + 2.43e4T^{2} \) |
| 31 | \( 1 + 188T + 2.97e4T^{2} \) |
| 37 | \( 1 - 133T + 5.06e4T^{2} \) |
| 41 | \( 1 + 36.3T + 6.89e4T^{2} \) |
| 43 | \( 1 - 72.7T + 7.95e4T^{2} \) |
| 47 | \( 1 - 72T + 1.03e5T^{2} \) |
| 53 | \( 1 + 45T + 1.48e5T^{2} \) |
| 59 | \( 1 - 378T + 2.05e5T^{2} \) |
| 61 | \( 1 + 623.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 386T + 3.00e5T^{2} \) |
| 71 | \( 1 + 198T + 3.57e5T^{2} \) |
| 73 | \( 1 + 76.2T + 3.89e5T^{2} \) |
| 79 | \( 1 - 152.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.24e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 45T + 7.04e5T^{2} \) |
| 97 | \( 1 - 89T + 9.12e5T^{2} \) |
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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
−11.41505408062244613497981974528, −10.56939826390575742678670119827, −9.386983654753553971010744110187, −7.61318835890389105509109530079, −6.89723074268839939536178584405, −5.66920484391210539756562407247, −5.08815290414079960065966673380, −4.39417990513750377918682885445, −2.71981149544223796803739153556, −1.66267948654698761608896529334,
1.66267948654698761608896529334, 2.71981149544223796803739153556, 4.39417990513750377918682885445, 5.08815290414079960065966673380, 5.66920484391210539756562407247, 6.89723074268839939536178584405, 7.61318835890389105509109530079, 9.386983654753553971010744110187, 10.56939826390575742678670119827, 11.41505408062244613497981974528
