| L(s) = 1 | + 5·2-s + 3·3-s + 17·4-s − 5·5-s + 15·6-s + 45·8-s + 9·9-s − 25·10-s + 12·11-s + 51·12-s − 30·13-s − 15·15-s + 89·16-s + 134·17-s + 45·18-s + 92·19-s − 85·20-s + 60·22-s + 112·23-s + 135·24-s + 25·25-s − 150·26-s + 27·27-s − 58·29-s − 75·30-s + 224·31-s + 85·32-s + ⋯ |
| L(s) = 1 | + 1.76·2-s + 0.577·3-s + 17/8·4-s − 0.447·5-s + 1.02·6-s + 1.98·8-s + 1/3·9-s − 0.790·10-s + 0.328·11-s + 1.22·12-s − 0.640·13-s − 0.258·15-s + 1.39·16-s + 1.91·17-s + 0.589·18-s + 1.11·19-s − 0.950·20-s + 0.581·22-s + 1.01·23-s + 1.14·24-s + 1/5·25-s − 1.13·26-s + 0.192·27-s − 0.371·29-s − 0.456·30-s + 1.29·31-s + 0.469·32-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
| \(L(2)\) |
\(\approx\) |
\(7.585118889\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.585118889\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 5 T + p^{3} T^{2} \) |
| 11 | \( 1 - 12 T + p^{3} T^{2} \) |
| 13 | \( 1 + 30 T + p^{3} T^{2} \) |
| 17 | \( 1 - 134 T + p^{3} T^{2} \) |
| 19 | \( 1 - 92 T + p^{3} T^{2} \) |
| 23 | \( 1 - 112 T + p^{3} T^{2} \) |
| 29 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 31 | \( 1 - 224 T + p^{3} T^{2} \) |
| 37 | \( 1 + 146 T + p^{3} T^{2} \) |
| 41 | \( 1 + 18 T + p^{3} T^{2} \) |
| 43 | \( 1 - 340 T + p^{3} T^{2} \) |
| 47 | \( 1 + 208 T + p^{3} T^{2} \) |
| 53 | \( 1 + 754 T + p^{3} T^{2} \) |
| 59 | \( 1 + 380 T + p^{3} T^{2} \) |
| 61 | \( 1 + 718 T + p^{3} T^{2} \) |
| 67 | \( 1 - 412 T + p^{3} T^{2} \) |
| 71 | \( 1 + 960 T + p^{3} T^{2} \) |
| 73 | \( 1 + 1066 T + p^{3} T^{2} \) |
| 79 | \( 1 - 896 T + p^{3} T^{2} \) |
| 83 | \( 1 + 436 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1038 T + p^{3} T^{2} \) |
| 97 | \( 1 - 702 T + p^{3} T^{2} \) |
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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
−10.13782493954537041936852537377, −9.198840373065645582279772371736, −7.78606683291345362664398392121, −7.34010908234651063693470824190, −6.23742976955526156172638730779, −5.23983388549933116520512280298, −4.50436749000921993551687039077, −3.34606429040730818303481939033, −2.94180167768399390459546265050, −1.36050018965851097692471491156,
1.36050018965851097692471491156, 2.94180167768399390459546265050, 3.34606429040730818303481939033, 4.50436749000921993551687039077, 5.23983388549933116520512280298, 6.23742976955526156172638730779, 7.34010908234651063693470824190, 7.78606683291345362664398392121, 9.198840373065645582279772371736, 10.13782493954537041936852537377
