| L(s) = 1 | + 5.37·3-s + 11.8·5-s + 26.4·7-s + 1.86·9-s + 49.8·11-s − 49.0·13-s + 63.7·15-s + 17.2·17-s + 19·19-s + 142.·21-s − 166.·23-s + 15.6·25-s − 135.·27-s − 109.·29-s + 273.·31-s + 267.·33-s + 314.·35-s + 167.·37-s − 263.·39-s + 15.1·41-s − 413.·43-s + 22.0·45-s + 161.·47-s + 358.·49-s + 92.5·51-s − 490.·53-s + 591.·55-s + ⋯ |
| L(s) = 1 | + 1.03·3-s + 1.06·5-s + 1.43·7-s + 0.0689·9-s + 1.36·11-s − 1.04·13-s + 1.09·15-s + 0.245·17-s + 0.229·19-s + 1.47·21-s − 1.51·23-s + 0.125·25-s − 0.962·27-s − 0.699·29-s + 1.58·31-s + 1.41·33-s + 1.51·35-s + 0.742·37-s − 1.08·39-s + 0.0577·41-s − 1.46·43-s + 0.0731·45-s + 0.501·47-s + 1.04·49-s + 0.254·51-s − 1.27·53-s + 1.44·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.559730687\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.559730687\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - 19T \) |
| good | 3 | \( 1 - 5.37T + 27T^{2} \) |
| 5 | \( 1 - 11.8T + 125T^{2} \) |
| 7 | \( 1 - 26.4T + 343T^{2} \) |
| 11 | \( 1 - 49.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 49.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 17.2T + 4.91e3T^{2} \) |
| 23 | \( 1 + 166.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 109.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 273.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 167.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 15.1T + 6.89e4T^{2} \) |
| 43 | \( 1 + 413.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 161.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 490.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 335.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 725.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 497.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 798.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 311.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 665.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 372.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 673.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 960.T + 9.12e5T^{2} \) |
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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
−11.44074591378052331103512113893, −9.987638382394648266856100206738, −9.453970025955761482759052546352, −8.409857930690628276882263752098, −7.72315221331815903144200362301, −6.35891654005827199974563264306, −5.20387394885263760812976044008, −3.99665700438603626216575014331, −2.41965971956718394396818241026, −1.56850168219061567845452618242,
1.56850168219061567845452618242, 2.41965971956718394396818241026, 3.99665700438603626216575014331, 5.20387394885263760812976044008, 6.35891654005827199974563264306, 7.72315221331815903144200362301, 8.409857930690628276882263752098, 9.453970025955761482759052546352, 9.987638382394648266856100206738, 11.44074591378052331103512113893
