| L(s) = 1 | − 1.16·3-s − 1.28·5-s − 30.0·7-s − 25.6·9-s + 7.58·11-s − 15.2·13-s + 1.50·15-s − 17·17-s + 41.7·19-s + 35.1·21-s − 84.4·23-s − 123.·25-s + 61.5·27-s − 155.·29-s − 217.·31-s − 8.86·33-s + 38.6·35-s − 370.·37-s + 17.7·39-s + 97.9·41-s + 413.·43-s + 32.9·45-s + 393.·47-s + 559.·49-s + 19.8·51-s + 245.·53-s − 9.75·55-s + ⋯ |
| L(s) = 1 | − 0.224·3-s − 0.114·5-s − 1.62·7-s − 0.949·9-s + 0.207·11-s − 0.324·13-s + 0.0258·15-s − 0.242·17-s + 0.504·19-s + 0.364·21-s − 0.765·23-s − 0.986·25-s + 0.438·27-s − 0.994·29-s − 1.25·31-s − 0.0467·33-s + 0.186·35-s − 1.64·37-s + 0.0729·39-s + 0.373·41-s + 1.46·43-s + 0.109·45-s + 1.22·47-s + 1.63·49-s + 0.0545·51-s + 0.637·53-s − 0.0239·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.5648174696\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5648174696\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + 17T \) |
| good | 3 | \( 1 + 1.16T + 27T^{2} \) |
| 5 | \( 1 + 1.28T + 125T^{2} \) |
| 7 | \( 1 + 30.0T + 343T^{2} \) |
| 11 | \( 1 - 7.58T + 1.33e3T^{2} \) |
| 13 | \( 1 + 15.2T + 2.19e3T^{2} \) |
| 19 | \( 1 - 41.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 84.4T + 1.21e4T^{2} \) |
| 29 | \( 1 + 155.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 217.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 370.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 97.9T + 6.89e4T^{2} \) |
| 43 | \( 1 - 413.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 393.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 245.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 183.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 214.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 727.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 431.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 540.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 511.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 834.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 495.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 892.T + 9.12e5T^{2} \) |
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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
−9.353140243666380939614154977267, −8.965928702532280441599322286554, −7.69951282681870649788151906835, −6.95871254493104344743247180399, −5.96415808146488500148303846523, −5.52632777571192169554157262035, −4.00211365313484793043893378856, −3.28001326430056134396016047390, −2.19276303083167217665971990897, −0.37166746258032935007591157366,
0.37166746258032935007591157366, 2.19276303083167217665971990897, 3.28001326430056134396016047390, 4.00211365313484793043893378856, 5.52632777571192169554157262035, 5.96415808146488500148303846523, 6.95871254493104344743247180399, 7.69951282681870649788151906835, 8.965928702532280441599322286554, 9.353140243666380939614154977267
