| L(s) = 1 | + 2·2-s + 4·4-s − 15.2·5-s + 8·8-s − 30.4·10-s + 2·11-s + 30.4·13-s + 16·16-s + 45.6·17-s − 152.·19-s − 60.9·20-s + 4·22-s + 30·23-s + 107.·25-s + 60.9·26-s + 212·29-s − 213.·31-s + 32·32-s + 91.3·34-s + 246·37-s − 304.·38-s − 121.·40-s + 319.·41-s − 284·43-s + 8·44-s + 60·46-s − 60.9·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.36·5-s + 0.353·8-s − 0.963·10-s + 0.0548·11-s + 0.649·13-s + 0.250·16-s + 0.651·17-s − 1.83·19-s − 0.681·20-s + 0.0387·22-s + 0.271·23-s + 0.856·25-s + 0.459·26-s + 1.35·29-s − 1.23·31-s + 0.176·32-s + 0.460·34-s + 1.09·37-s − 1.30·38-s − 0.481·40-s + 1.21·41-s − 1.00·43-s + 0.0274·44-s + 0.192·46-s − 0.189·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.396433019\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.396433019\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 15.2T + 125T^{2} \) |
| 11 | \( 1 - 2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 30.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 45.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 152.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 30T + 1.21e4T^{2} \) |
| 29 | \( 1 - 212T + 2.43e4T^{2} \) |
| 31 | \( 1 + 213.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 246T + 5.06e4T^{2} \) |
| 41 | \( 1 - 319.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 284T + 7.95e4T^{2} \) |
| 47 | \( 1 + 60.9T + 1.03e5T^{2} \) |
| 53 | \( 1 - 548T + 1.48e5T^{2} \) |
| 59 | \( 1 - 670.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 517.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 652T + 3.00e5T^{2} \) |
| 71 | \( 1 - 770T + 3.57e5T^{2} \) |
| 73 | \( 1 - 974.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 472T + 4.93e5T^{2} \) |
| 83 | \( 1 + 182.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 715.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 304.T + 9.12e5T^{2} \) |
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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
−9.907401778901659840574087095881, −8.563814413044450468794385375770, −8.106838637129102104146975865514, −7.08472205645968474015186744769, −6.33188242912368712903540146598, −5.20954825451139873746524539571, −4.13393449563457430196788292015, −3.67554398276607492804530571330, −2.39340057781635065634602404917, −0.76072522242200841928762099869,
0.76072522242200841928762099869, 2.39340057781635065634602404917, 3.67554398276607492804530571330, 4.13393449563457430196788292015, 5.20954825451139873746524539571, 6.33188242912368712903540146598, 7.08472205645968474015186744769, 8.106838637129102104146975865514, 8.563814413044450468794385375770, 9.907401778901659840574087095881
