| L(s) = 1 | + 9.70·3-s + 5·5-s + 7·7-s + 67.2·9-s + 48.7·11-s − 6.26·13-s + 48.5·15-s + 24.6·17-s − 35.8·19-s + 67.9·21-s − 17.5·23-s + 25·25-s + 390.·27-s − 186.·29-s − 95.8·31-s + 473.·33-s + 35·35-s + 345.·37-s − 60.7·39-s − 284.·41-s + 351.·43-s + 336.·45-s + 161.·47-s + 49·49-s + 239.·51-s + 605.·53-s + 243.·55-s + ⋯ |
| L(s) = 1 | + 1.86·3-s + 0.447·5-s + 0.377·7-s + 2.48·9-s + 1.33·11-s − 0.133·13-s + 0.835·15-s + 0.352·17-s − 0.432·19-s + 0.706·21-s − 0.159·23-s + 0.200·25-s + 2.78·27-s − 1.19·29-s − 0.555·31-s + 2.49·33-s + 0.169·35-s + 1.53·37-s − 0.249·39-s − 1.08·41-s + 1.24·43-s + 1.11·45-s + 0.501·47-s + 0.142·49-s + 0.657·51-s + 1.56·53-s + 0.597·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.514073895\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.514073895\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 7 | \( 1 - 7T \) |
| good | 3 | \( 1 - 9.70T + 27T^{2} \) |
| 11 | \( 1 - 48.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 6.26T + 2.19e3T^{2} \) |
| 17 | \( 1 - 24.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 35.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 17.5T + 1.21e4T^{2} \) |
| 29 | \( 1 + 186.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 95.8T + 2.97e4T^{2} \) |
| 37 | \( 1 - 345.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 284.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 351.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 161.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 605.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 175.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 362.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 751.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 827.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 963.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 677.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 811.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.03e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.15e3T + 9.12e5T^{2} \) |
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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
−8.801878297721115456587246431589, −8.015341311500079421239758845769, −7.33524150179279619512080692607, −6.59730626951983627455666766914, −5.50497898688341102149375255032, −4.20756637733645269592825416710, −3.83325075322952957514986421987, −2.73679017425003513551641802479, −1.95505299004343207114257205203, −1.16171584569672250713661498234,
1.16171584569672250713661498234, 1.95505299004343207114257205203, 2.73679017425003513551641802479, 3.83325075322952957514986421987, 4.20756637733645269592825416710, 5.50497898688341102149375255032, 6.59730626951983627455666766914, 7.33524150179279619512080692607, 8.015341311500079421239758845769, 8.801878297721115456587246431589
