| L(s) = 1 | + 2-s − 3·3-s − 7·4-s − 5·5-s − 3·6-s − 15·8-s + 9·9-s − 5·10-s + 52·11-s + 21·12-s − 22·13-s + 15·15-s + 41·16-s + 14·17-s + 9·18-s + 20·19-s + 35·20-s + 52·22-s − 168·23-s + 45·24-s + 25·25-s − 22·26-s − 27·27-s + 230·29-s + 15·30-s + 288·31-s + 161·32-s + ⋯ |
| L(s) = 1 | + 0.353·2-s − 0.577·3-s − 7/8·4-s − 0.447·5-s − 0.204·6-s − 0.662·8-s + 1/3·9-s − 0.158·10-s + 1.42·11-s + 0.505·12-s − 0.469·13-s + 0.258·15-s + 0.640·16-s + 0.199·17-s + 0.117·18-s + 0.241·19-s + 0.391·20-s + 0.503·22-s − 1.52·23-s + 0.382·24-s + 1/5·25-s − 0.165·26-s − 0.192·27-s + 1.47·29-s + 0.0912·30-s + 1.66·31-s + 0.889·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + p T \) |
| 5 | \( 1 + p T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - T + p^{3} T^{2} \) |
| 11 | \( 1 - 52 T + p^{3} T^{2} \) |
| 13 | \( 1 + 22 T + p^{3} T^{2} \) |
| 17 | \( 1 - 14 T + p^{3} T^{2} \) |
| 19 | \( 1 - 20 T + p^{3} T^{2} \) |
| 23 | \( 1 + 168 T + p^{3} T^{2} \) |
| 29 | \( 1 - 230 T + p^{3} T^{2} \) |
| 31 | \( 1 - 288 T + p^{3} T^{2} \) |
| 37 | \( 1 + 34 T + p^{3} T^{2} \) |
| 41 | \( 1 + 122 T + p^{3} T^{2} \) |
| 43 | \( 1 + 188 T + p^{3} T^{2} \) |
| 47 | \( 1 + 256 T + p^{3} T^{2} \) |
| 53 | \( 1 + 338 T + p^{3} T^{2} \) |
| 59 | \( 1 + 100 T + p^{3} T^{2} \) |
| 61 | \( 1 + 742 T + p^{3} T^{2} \) |
| 67 | \( 1 + 84 T + p^{3} T^{2} \) |
| 71 | \( 1 + 328 T + p^{3} T^{2} \) |
| 73 | \( 1 - 38 T + p^{3} T^{2} \) |
| 79 | \( 1 + 240 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1212 T + p^{3} T^{2} \) |
| 89 | \( 1 + 330 T + p^{3} T^{2} \) |
| 97 | \( 1 + 866 T + p^{3} T^{2} \) |
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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
−9.711312958583835315407542504733, −8.677596103774149744617904139337, −7.927135436645977516371813102783, −6.66803885179905124083649801266, −5.99608240035398561509404126765, −4.77422069887325687342669496230, −4.24091330990758756248946488495, −3.17059976559507133876237794322, −1.26721911484901003940594312255, 0,
1.26721911484901003940594312255, 3.17059976559507133876237794322, 4.24091330990758756248946488495, 4.77422069887325687342669496230, 5.99608240035398561509404126765, 6.66803885179905124083649801266, 7.927135436645977516371813102783, 8.677596103774149744617904139337, 9.711312958583835315407542504733
