| L(s) = 1 | + 3.40·2-s + 7.32·3-s + 3.61·4-s − 17.2·5-s + 24.9·6-s − 15.1·7-s − 14.9·8-s + 26.6·9-s − 58.9·10-s + 61.3·11-s + 26.4·12-s + 59.3·13-s − 51.6·14-s − 126.·15-s − 79.8·16-s + 62.2·17-s + 90.9·18-s − 71.9·19-s − 62.4·20-s − 111.·21-s + 209.·22-s − 39.7·23-s − 109.·24-s + 173.·25-s + 202.·26-s − 2.30·27-s − 54.8·28-s + ⋯ |
| L(s) = 1 | + 1.20·2-s + 1.41·3-s + 0.451·4-s − 1.54·5-s + 1.69·6-s − 0.818·7-s − 0.660·8-s + 0.988·9-s − 1.86·10-s + 1.68·11-s + 0.637·12-s + 1.26·13-s − 0.986·14-s − 2.18·15-s − 1.24·16-s + 0.888·17-s + 1.19·18-s − 0.868·19-s − 0.698·20-s − 1.15·21-s + 2.02·22-s − 0.360·23-s − 0.931·24-s + 1.39·25-s + 1.52·26-s − 0.0164·27-s − 0.369·28-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
| \(L(2)\) |
\(\approx\) |
\(2.414594560\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.414594560\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 3.40T + 8T^{2} \) |
| 3 | \( 1 - 7.32T + 27T^{2} \) |
| 5 | \( 1 + 17.2T + 125T^{2} \) |
| 7 | \( 1 + 15.1T + 343T^{2} \) |
| 11 | \( 1 - 61.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 59.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 62.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 71.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 39.7T + 1.21e4T^{2} \) |
| 29 | \( 1 + 132.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 59.1T + 2.97e4T^{2} \) |
| 41 | \( 1 - 146.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 369.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 101.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 244.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 522.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 682.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 235.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 965.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 665.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 82.1T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.29e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 24.4T + 7.04e5T^{2} \) |
| 97 | \( 1 + 676.T + 9.12e5T^{2} \) |
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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
−15.32633570248502596946489304475, −14.67441671039406503067420288280, −13.68790796920784886597234263001, −12.56425091371166917667321951982, −11.52992486868616710898258760073, −9.250110285241939407310010730642, −8.232983642415302491331342244425, −6.53556218705602231878724449322, −3.88352132058545875777878828197, −3.52737783387336051922432317502,
3.52737783387336051922432317502, 3.88352132058545875777878828197, 6.53556218705602231878724449322, 8.232983642415302491331342244425, 9.250110285241939407310010730642, 11.52992486868616710898258760073, 12.56425091371166917667321951982, 13.68790796920784886597234263001, 14.67441671039406503067420288280, 15.32633570248502596946489304475
