| L(s) = 1 | + 0.525·2-s − 7.72·4-s + 5·5-s − 8.26·8-s + 2.62·10-s + 48.9·11-s + 40.1·13-s + 57.4·16-s + 59.1·17-s + 129.·19-s − 38.6·20-s + 25.7·22-s + 93.5·23-s + 25·25-s + 21.1·26-s + 117.·29-s − 162.·31-s + 96.3·32-s + 31.0·34-s − 3.72·37-s + 68.0·38-s − 41.3·40-s − 222.·41-s − 62.2·43-s − 378.·44-s + 49.1·46-s − 460.·47-s + ⋯ |
| L(s) = 1 | + 0.185·2-s − 0.965·4-s + 0.447·5-s − 0.365·8-s + 0.0831·10-s + 1.34·11-s + 0.856·13-s + 0.897·16-s + 0.844·17-s + 1.56·19-s − 0.431·20-s + 0.249·22-s + 0.848·23-s + 0.200·25-s + 0.159·26-s + 0.750·29-s − 0.944·31-s + 0.532·32-s + 0.156·34-s − 0.0165·37-s + 0.290·38-s − 0.163·40-s − 0.848·41-s − 0.220·43-s − 1.29·44-s + 0.157·46-s − 1.42·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.808780106\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.808780106\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 0.525T + 8T^{2} \) |
| 11 | \( 1 - 48.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 40.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 59.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 129.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 93.5T + 1.21e4T^{2} \) |
| 29 | \( 1 - 117.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 162.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 3.72T + 5.06e4T^{2} \) |
| 41 | \( 1 + 222.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 62.2T + 7.95e4T^{2} \) |
| 47 | \( 1 + 460.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 492.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 153.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 194.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 667.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 256.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.21e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 381.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 281.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 926.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 634.T + 9.12e5T^{2} \) |
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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
−8.755182762497077139265916322154, −8.128138709215161015893669519330, −7.05365002504522057622242698329, −6.26081082502740276725722001569, −5.42465490507775679696645761672, −4.80509849354605235180280349966, −3.63959970941453307786083039591, −3.25099418534408643726125457154, −1.50934030570331313607083367000, −0.838726207795432649361489770264,
0.838726207795432649361489770264, 1.50934030570331313607083367000, 3.25099418534408643726125457154, 3.63959970941453307786083039591, 4.80509849354605235180280349966, 5.42465490507775679696645761672, 6.26081082502740276725722001569, 7.05365002504522057622242698329, 8.128138709215161015893669519330, 8.755182762497077139265916322154
