| L(s) = 1 | − 2.42·2-s + 0.381·3-s + 3.90·4-s − 2.17·5-s − 0.926·6-s − 2.70·7-s − 4.62·8-s − 2.85·9-s + 5.27·10-s + 5.11·11-s + 1.48·12-s + 6.54·13-s + 6.57·14-s − 0.828·15-s + 3.43·16-s − 0.0279·17-s + 6.93·18-s + 7.94·19-s − 8.47·20-s − 1.03·21-s − 12.4·22-s + 7.01·23-s − 1.76·24-s − 0.287·25-s − 15.8·26-s − 2.23·27-s − 10.5·28-s + ⋯ |
| L(s) = 1 | − 1.71·2-s + 0.220·3-s + 1.95·4-s − 0.970·5-s − 0.378·6-s − 1.02·7-s − 1.63·8-s − 0.951·9-s + 1.66·10-s + 1.54·11-s + 0.429·12-s + 1.81·13-s + 1.75·14-s − 0.213·15-s + 0.858·16-s − 0.00676·17-s + 1.63·18-s + 1.82·19-s − 1.89·20-s − 0.225·21-s − 2.64·22-s + 1.46·23-s − 0.360·24-s − 0.0575·25-s − 3.11·26-s − 0.429·27-s − 1.99·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6037 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6037 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8557246217\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8557246217\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 6037 | \( 1+O(T) \) |
| good | 2 | \( 1 + 2.42T + 2T^{2} \) |
| 3 | \( 1 - 0.381T + 3T^{2} \) |
| 5 | \( 1 + 2.17T + 5T^{2} \) |
| 7 | \( 1 + 2.70T + 7T^{2} \) |
| 11 | \( 1 - 5.11T + 11T^{2} \) |
| 13 | \( 1 - 6.54T + 13T^{2} \) |
| 17 | \( 1 + 0.0279T + 17T^{2} \) |
| 19 | \( 1 - 7.94T + 19T^{2} \) |
| 23 | \( 1 - 7.01T + 23T^{2} \) |
| 29 | \( 1 - 8.51T + 29T^{2} \) |
| 31 | \( 1 - 3.77T + 31T^{2} \) |
| 37 | \( 1 + 0.838T + 37T^{2} \) |
| 41 | \( 1 + 8.68T + 41T^{2} \) |
| 43 | \( 1 - 3.58T + 43T^{2} \) |
| 47 | \( 1 - 7.68T + 47T^{2} \) |
| 53 | \( 1 - 13.0T + 53T^{2} \) |
| 59 | \( 1 - 10.0T + 59T^{2} \) |
| 61 | \( 1 - 5.26T + 61T^{2} \) |
| 67 | \( 1 + 12.6T + 67T^{2} \) |
| 71 | \( 1 + 11.1T + 71T^{2} \) |
| 73 | \( 1 - 5.14T + 73T^{2} \) |
| 79 | \( 1 + 9.63T + 79T^{2} \) |
| 83 | \( 1 - 9.70T + 83T^{2} \) |
| 89 | \( 1 - 0.517T + 89T^{2} \) |
| 97 | \( 1 + 6.38T + 97T^{2} \) |
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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
−8.359104302703603416647165862378, −7.53163714552717254034438704027, −6.80959610437060196726696980721, −6.42100211414685280790403746512, −5.53243157974621790894230810913, −4.06848168290815826608263720965, −3.35221979977273089660766408000, −2.82393211104702696220365106787, −1.23376871908811858952722310522, −0.75019741962924613841098308396,
0.75019741962924613841098308396, 1.23376871908811858952722310522, 2.82393211104702696220365106787, 3.35221979977273089660766408000, 4.06848168290815826608263720965, 5.53243157974621790894230810913, 6.42100211414685280790403746512, 6.80959610437060196726696980721, 7.53163714552717254034438704027, 8.359104302703603416647165862378
