| L(s) = 1 | + 7.36·3-s − 15.4·5-s + 7·7-s + 27.2·9-s − 11·11-s + 49.0·13-s − 114.·15-s − 34.1·17-s + 144.·19-s + 51.5·21-s − 118.·23-s + 115.·25-s + 1.63·27-s − 63.6·29-s + 212.·31-s − 80.9·33-s − 108.·35-s − 200.·37-s + 361.·39-s + 451.·41-s + 130.·43-s − 421.·45-s + 176.·47-s + 49·49-s − 251.·51-s − 629.·53-s + 170.·55-s + ⋯ |
| L(s) = 1 | + 1.41·3-s − 1.38·5-s + 0.377·7-s + 1.00·9-s − 0.301·11-s + 1.04·13-s − 1.96·15-s − 0.486·17-s + 1.74·19-s + 0.535·21-s − 1.07·23-s + 0.920·25-s + 0.0116·27-s − 0.407·29-s + 1.23·31-s − 0.427·33-s − 0.523·35-s − 0.889·37-s + 1.48·39-s + 1.71·41-s + 0.463·43-s − 1.39·45-s + 0.547·47-s + 0.142·49-s − 0.689·51-s − 1.63·53-s + 0.417·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.961620500\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.961620500\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| 11 | \( 1 + 11T \) |
| good | 3 | \( 1 - 7.36T + 27T^{2} \) |
| 5 | \( 1 + 15.4T + 125T^{2} \) |
| 13 | \( 1 - 49.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 34.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 144.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 118.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 63.6T + 2.43e4T^{2} \) |
| 31 | \( 1 - 212.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 200.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 451.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 130.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 176.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 629.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 86.9T + 2.05e5T^{2} \) |
| 61 | \( 1 - 644.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 400.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 507.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 176.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 701.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.25e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 788.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 185.T + 9.12e5T^{2} \) |
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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
−9.084103149187224332932155599140, −8.414162923128295552239366220701, −7.78340063696898166696936784398, −7.40828808701396361694788376280, −6.07633075650367756801034945215, −4.77289357626319325068913595923, −3.81530721510154809140877502706, −3.33365835942593853335432881803, −2.20755400427456565281873324598, −0.830574608514208413296298699731,
0.830574608514208413296298699731, 2.20755400427456565281873324598, 3.33365835942593853335432881803, 3.81530721510154809140877502706, 4.77289357626319325068913595923, 6.07633075650367756801034945215, 7.40828808701396361694788376280, 7.78340063696898166696936784398, 8.414162923128295552239366220701, 9.084103149187224332932155599140
