| L(s) = 1 | − 2.56·2-s + 9.45·3-s − 1.40·4-s + 1.50·5-s − 24.2·6-s + 12.6·7-s + 24.1·8-s + 62.3·9-s − 3.87·10-s − 3.77·11-s − 13.3·12-s − 71.0·13-s − 32.5·14-s + 14.2·15-s − 50.7·16-s + 38.8·17-s − 160.·18-s − 64.4·19-s − 2.12·20-s + 119.·21-s + 9.68·22-s − 197.·23-s + 228.·24-s − 122.·25-s + 182.·26-s + 334.·27-s − 17.8·28-s + ⋯ |
| L(s) = 1 | − 0.907·2-s + 1.81·3-s − 0.175·4-s + 0.134·5-s − 1.65·6-s + 0.685·7-s + 1.06·8-s + 2.30·9-s − 0.122·10-s − 0.103·11-s − 0.319·12-s − 1.51·13-s − 0.621·14-s + 0.245·15-s − 0.793·16-s + 0.553·17-s − 2.09·18-s − 0.778·19-s − 0.0237·20-s + 1.24·21-s + 0.0938·22-s − 1.79·23-s + 1.94·24-s − 0.981·25-s + 1.37·26-s + 2.38·27-s − 0.120·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.313171386\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.313171386\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + 37T \) |
| good | 2 | \( 1 + 2.56T + 8T^{2} \) |
| 3 | \( 1 - 9.45T + 27T^{2} \) |
| 5 | \( 1 - 1.50T + 125T^{2} \) |
| 7 | \( 1 - 12.6T + 343T^{2} \) |
| 11 | \( 1 + 3.77T + 1.33e3T^{2} \) |
| 13 | \( 1 + 71.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 38.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 64.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 197.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 255.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 97.3T + 2.97e4T^{2} \) |
| 41 | \( 1 - 350.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 138.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 305.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 310.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 112.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 595.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 899.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 450.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 469.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 263.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 314.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 772.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 190.T + 9.12e5T^{2} \) |
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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
−15.79610655733135980958509262879, −14.41685936137979562723149706310, −13.99528873457439362350771372074, −12.50572402022310300989322839318, −10.23462440308610451195877494840, −9.448568830873263343152692125698, −8.221993368651064041117971735082, −7.59817395508620864325194153323, −4.35705744866897044148136680398, −2.12383503890829448781329682417,
2.12383503890829448781329682417, 4.35705744866897044148136680398, 7.59817395508620864325194153323, 8.221993368651064041117971735082, 9.448568830873263343152692125698, 10.23462440308610451195877494840, 12.50572402022310300989322839318, 13.99528873457439362350771372074, 14.41685936137979562723149706310, 15.79610655733135980958509262879
