| L(s) = 1 | + 3·3-s + 21.3·5-s + 9·9-s + 31.3·11-s − 84.7·13-s + 64.0·15-s + 16.0·17-s − 20·19-s − 80.7·23-s + 331.·25-s + 27·27-s + 102·29-s + 245.·31-s + 94.0·33-s + 215.·37-s − 254.·39-s − 150.·41-s + 441.·43-s + 192.·45-s + 206.·47-s + 48.2·51-s + 426.·53-s + 669.·55-s − 60·57-s − 363.·59-s + 343.·61-s − 1.80e3·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.91·5-s + 0.333·9-s + 0.859·11-s − 1.80·13-s + 1.10·15-s + 0.229·17-s − 0.241·19-s − 0.732·23-s + 2.64·25-s + 0.192·27-s + 0.653·29-s + 1.42·31-s + 0.496·33-s + 0.956·37-s − 1.04·39-s − 0.574·41-s + 1.56·43-s + 0.636·45-s + 0.640·47-s + 0.132·51-s + 1.10·53-s + 1.64·55-s − 0.139·57-s − 0.801·59-s + 0.721·61-s − 3.45·65-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
| \(L(2)\) |
\(\approx\) |
\(4.091088126\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.091088126\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 21.3T + 125T^{2} \) |
| 11 | \( 1 - 31.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 84.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 16.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 20T + 6.85e3T^{2} \) |
| 23 | \( 1 + 80.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 102T + 2.43e4T^{2} \) |
| 31 | \( 1 - 245.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 215.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 150.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 441.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 206.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 426.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 363.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 343.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 69.2T + 3.00e5T^{2} \) |
| 71 | \( 1 + 468.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 747.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.29e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.29e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 563.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.48e3T + 9.12e5T^{2} \) |
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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
−9.509599816015690414425847846568, −8.847035982021804425222343869864, −7.76170822180029035495302287594, −6.77591550613549412981129215370, −6.11981081410916523755818810394, −5.16724276032225045010682939807, −4.27814335858965591242735782051, −2.71780340321345677952077662760, −2.22789930601882573389399924082, −1.07238180131071605084645991421,
1.07238180131071605084645991421, 2.22789930601882573389399924082, 2.71780340321345677952077662760, 4.27814335858965591242735782051, 5.16724276032225045010682939807, 6.11981081410916523755818810394, 6.77591550613549412981129215370, 7.76170822180029035495302287594, 8.847035982021804425222343869864, 9.509599816015690414425847846568
