| L(s) = 1 | − 3.70·2-s − 5.70·3-s + 5.70·4-s + 21.1·6-s + 8.50·8-s + 5.50·9-s − 60.0·11-s − 32.5·12-s + 0.387·13-s − 77.1·16-s − 35.4·17-s − 20.3·18-s + 6.08·19-s + 222.·22-s + 31.5·23-s − 48.5·24-s − 1.43·26-s + 122.·27-s − 292.·29-s − 130.·31-s + 217.·32-s + 342.·33-s + 131.·34-s + 31.4·36-s − 219.·37-s − 22.5·38-s − 2.20·39-s + ⋯ |
| L(s) = 1 | − 1.30·2-s − 1.09·3-s + 0.712·4-s + 1.43·6-s + 0.375·8-s + 0.203·9-s − 1.64·11-s − 0.782·12-s + 0.00826·13-s − 1.20·16-s − 0.506·17-s − 0.266·18-s + 0.0735·19-s + 2.15·22-s + 0.285·23-s − 0.412·24-s − 0.0108·26-s + 0.873·27-s − 1.87·29-s − 0.754·31-s + 1.20·32-s + 1.80·33-s + 0.662·34-s + 0.145·36-s − 0.977·37-s − 0.0962·38-s − 0.00907·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.05351258468\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05351258468\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 3.70T + 8T^{2} \) |
| 3 | \( 1 + 5.70T + 27T^{2} \) |
| 11 | \( 1 + 60.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 0.387T + 2.19e3T^{2} \) |
| 17 | \( 1 + 35.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 6.08T + 6.85e3T^{2} \) |
| 23 | \( 1 - 31.5T + 1.21e4T^{2} \) |
| 29 | \( 1 + 292.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 130.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 219.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 447.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 210.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 457.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 144.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 767.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 667.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 77.4T + 3.00e5T^{2} \) |
| 71 | \( 1 + 906.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.02e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 690.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 979.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 910.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 11.1T + 9.12e5T^{2} \) |
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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
−9.312144710886985983574344385352, −8.672994436315380494406483350516, −7.62828586202697288559249550573, −7.24039175612004127202160946284, −5.98425370458440510995349275089, −5.32348790126992751945559774953, −4.39545732124155146864463272088, −2.77635314221715415030863428589, −1.56480391480945291709115318272, −0.15318667703053597618608718381,
0.15318667703053597618608718381, 1.56480391480945291709115318272, 2.77635314221715415030863428589, 4.39545732124155146864463272088, 5.32348790126992751945559774953, 5.98425370458440510995349275089, 7.24039175612004127202160946284, 7.62828586202697288559249550573, 8.672994436315380494406483350516, 9.312144710886985983574344385352
