| L(s) = 1 | − 4.80·2-s + 3·3-s + 15.1·4-s + 5·5-s − 14.4·6-s − 34.1·8-s + 9·9-s − 24.0·10-s + 53.2·11-s + 45.3·12-s + 60.9·13-s + 15·15-s + 43.2·16-s + 31.1·17-s − 43.2·18-s − 17.9·19-s + 75.5·20-s − 255.·22-s − 34.8·23-s − 102.·24-s + 25·25-s − 292.·26-s + 27·27-s + 141.·29-s − 72.0·30-s + 117.·31-s + 65.1·32-s + ⋯ |
| L(s) = 1 | − 1.69·2-s + 0.577·3-s + 1.88·4-s + 0.447·5-s − 0.981·6-s − 1.50·8-s + 0.333·9-s − 0.759·10-s + 1.45·11-s + 1.08·12-s + 1.30·13-s + 0.258·15-s + 0.675·16-s + 0.443·17-s − 0.566·18-s − 0.216·19-s + 0.844·20-s − 2.48·22-s − 0.315·23-s − 0.871·24-s + 0.200·25-s − 2.20·26-s + 0.192·27-s + 0.905·29-s − 0.438·30-s + 0.682·31-s + 0.360·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.536950443\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.536950443\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 5 | \( 1 - 5T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 4.80T + 8T^{2} \) |
| 11 | \( 1 - 53.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 60.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 31.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 17.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 34.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 141.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 117.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 175.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 411.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 498.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 290.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 582.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 657.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 417.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 567.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 887.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.09e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 135.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 464.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 31.8T + 7.04e5T^{2} \) |
| 97 | \( 1 + 254.T + 9.12e5T^{2} \) |
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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
−9.780433935760052100696124456133, −9.049578756549598300935307921827, −8.529018963271609250773449045931, −7.73440268675887245164458385735, −6.63557797302969230360466167108, −6.12141400302357032994485880994, −4.27530223112602512493758069265, −3.02662306448184205829162452995, −1.70016164869208188515200772240, −0.981736770778698454641450064248,
0.981736770778698454641450064248, 1.70016164869208188515200772240, 3.02662306448184205829162452995, 4.27530223112602512493758069265, 6.12141400302357032994485880994, 6.63557797302969230360466167108, 7.73440268675887245164458385735, 8.529018963271609250773449045931, 9.049578756549598300935307921827, 9.780433935760052100696124456133
