| L(s) = 1 | + 3.46·2-s + 3·3-s + 3.99·4-s + 5·5-s + 10.3·6-s + 31.1·7-s − 13.8·8-s + 9·9-s + 17.3·10-s + 11.9·12-s − 76.2·13-s + 108·14-s + 15·15-s − 80·16-s + 46.7·17-s + 31.1·18-s − 31.1·19-s + 19.9·20-s + 93.5·21-s + 195·23-s − 41.5·24-s + 25·25-s − 264·26-s + 27·27-s + 124.·28-s + 34.6·29-s + 51.9·30-s + ⋯ |
| L(s) = 1 | + 1.22·2-s + 0.577·3-s + 0.499·4-s + 0.447·5-s + 0.707·6-s + 1.68·7-s − 0.612·8-s + 0.333·9-s + 0.547·10-s + 0.288·12-s − 1.62·13-s + 2.06·14-s + 0.258·15-s − 1.25·16-s + 0.667·17-s + 0.408·18-s − 0.376·19-s + 0.223·20-s + 0.971·21-s + 1.76·23-s − 0.353·24-s + 0.200·25-s − 1.99·26-s + 0.192·27-s + 0.841·28-s + 0.221·29-s + 0.316·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.731441860\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.731441860\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 5 | \( 1 - 5T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 3.46T + 8T^{2} \) |
| 7 | \( 1 - 31.1T + 343T^{2} \) |
| 13 | \( 1 + 76.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 46.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 31.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 195T + 1.21e4T^{2} \) |
| 29 | \( 1 - 34.6T + 2.43e4T^{2} \) |
| 31 | \( 1 - 97T + 2.97e4T^{2} \) |
| 37 | \( 1 - 274T + 5.06e4T^{2} \) |
| 41 | \( 1 - 384.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 415.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 3T + 1.03e5T^{2} \) |
| 53 | \( 1 - 3T + 1.48e5T^{2} \) |
| 59 | \( 1 + 648T + 2.05e5T^{2} \) |
| 61 | \( 1 + 458.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 70T + 3.00e5T^{2} \) |
| 71 | \( 1 + 372T + 3.57e5T^{2} \) |
| 73 | \( 1 - 935.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 521.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 45.0T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.12e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 470T + 9.12e5T^{2} \) |
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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
−9.020415492340436091207147850099, −7.898330417445382248980147621505, −7.43230277029857419057003768353, −6.30803203395439759784562414877, −5.28621874690842429323146325444, −4.80757806477824896988595153503, −4.20375464771236087929042708642, −2.85085708624874405759539496436, −2.32741617231247832729762673611, −1.03848987213083492164566398181,
1.03848987213083492164566398181, 2.32741617231247832729762673611, 2.85085708624874405759539496436, 4.20375464771236087929042708642, 4.80757806477824896988595153503, 5.28621874690842429323146325444, 6.30803203395439759784562414877, 7.43230277029857419057003768353, 7.898330417445382248980147621505, 9.020415492340436091207147850099
