| L(s) = 1 | + 3·3-s + 18.9·5-s + 9·9-s + 54.7·11-s + 62.0·13-s + 56.8·15-s + 122.·17-s − 12.5·19-s − 74.4·23-s + 234.·25-s + 27·27-s − 232.·29-s + 10.3·31-s + 164.·33-s − 245.·37-s + 186.·39-s + 238.·41-s − 92.9·43-s + 170.·45-s − 485.·47-s + 367.·51-s − 378.·53-s + 1.03e3·55-s − 37.5·57-s − 182.·59-s + 396.·61-s + 1.17e3·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.69·5-s + 0.333·9-s + 1.50·11-s + 1.32·13-s + 0.978·15-s + 1.74·17-s − 0.151·19-s − 0.674·23-s + 1.87·25-s + 0.192·27-s − 1.48·29-s + 0.0600·31-s + 0.866·33-s − 1.09·37-s + 0.763·39-s + 0.909·41-s − 0.329·43-s + 0.565·45-s − 1.50·47-s + 1.00·51-s − 0.981·53-s + 2.54·55-s − 0.0871·57-s − 0.403·59-s + 0.832·61-s + 2.24·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.713791838\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.713791838\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 18.9T + 125T^{2} \) |
| 11 | \( 1 - 54.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 62.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 122.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 12.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 74.4T + 1.21e4T^{2} \) |
| 29 | \( 1 + 232.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 10.3T + 2.97e4T^{2} \) |
| 37 | \( 1 + 245.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 238.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 92.9T + 7.95e4T^{2} \) |
| 47 | \( 1 + 485.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 378.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 182.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 396.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 261.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 874.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 152.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 573.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 317.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 95.0T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.60e3T + 9.12e5T^{2} \) |
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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
−9.428239210058783853153994189599, −8.808652523682976061794742834546, −7.87421838288676958874148594619, −6.69817152762523770827340526192, −6.06225864536583420406902009613, −5.37768517367236021068326948025, −3.95377323180451324122248407075, −3.17746322834528884210082672283, −1.75222236242182450846237818155, −1.31905381583327072366197586594,
1.31905381583327072366197586594, 1.75222236242182450846237818155, 3.17746322834528884210082672283, 3.95377323180451324122248407075, 5.37768517367236021068326948025, 6.06225864536583420406902009613, 6.69817152762523770827340526192, 7.87421838288676958874148594619, 8.808652523682976061794742834546, 9.428239210058783853153994189599
