| L(s) = 1 | − 9.02·3-s − 5·5-s − 4.33·7-s + 54.4·9-s − 48.1·11-s + 53.0·13-s + 45.1·15-s − 61.0·17-s − 12.0·19-s + 39.1·21-s + 23·23-s + 25·25-s − 247.·27-s + 198.·29-s + 11.9·31-s + 434.·33-s + 21.6·35-s − 230.·37-s − 478.·39-s − 393.·41-s + 167.·43-s − 272.·45-s − 531.·47-s − 324.·49-s + 550.·51-s + 373.·53-s + 240.·55-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 0.447·5-s − 0.234·7-s + 2.01·9-s − 1.32·11-s + 1.13·13-s + 0.776·15-s − 0.870·17-s − 0.145·19-s + 0.406·21-s + 0.208·23-s + 0.200·25-s − 1.76·27-s + 1.26·29-s + 0.0693·31-s + 2.29·33-s + 0.104·35-s − 1.02·37-s − 1.96·39-s − 1.49·41-s + 0.594·43-s − 0.901·45-s − 1.64·47-s − 0.945·49-s + 1.51·51-s + 0.968·53-s + 0.590·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3862566225\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3862566225\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| 23 | \( 1 - 23T \) |
| good | 3 | \( 1 + 9.02T + 27T^{2} \) |
| 7 | \( 1 + 4.33T + 343T^{2} \) |
| 11 | \( 1 + 48.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 53.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 61.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 12.0T + 6.85e3T^{2} \) |
| 29 | \( 1 - 198.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 11.9T + 2.97e4T^{2} \) |
| 37 | \( 1 + 230.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 393.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 167.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 531.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 373.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 825.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 632.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 38.7T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.10e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 446.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 642.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 292.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 273.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 959.T + 9.12e5T^{2} \) |
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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
−8.781446522657588053170262112527, −8.081533419461652587691825655646, −6.97212813779038954029199928633, −6.51526448523464857895964599806, −5.63233902932405453909453151221, −4.96244245379010999241593415194, −4.20406930021661692280050211869, −3.01581897869168559924565037144, −1.50843531302146611072390075804, −0.32965357354902561031073319266,
0.32965357354902561031073319266, 1.50843531302146611072390075804, 3.01581897869168559924565037144, 4.20406930021661692280050211869, 4.96244245379010999241593415194, 5.63233902932405453909453151221, 6.51526448523464857895964599806, 6.97212813779038954029199928633, 8.081533419461652587691825655646, 8.781446522657588053170262112527
