L(s) = 1 | − 2.60·2-s − 2.50·3-s + 4.77·4-s + 0.421·5-s + 6.52·6-s − 1.06·7-s − 7.20·8-s + 3.29·9-s − 1.09·10-s − 2.57·11-s − 11.9·12-s + 3.70·13-s + 2.76·14-s − 1.05·15-s + 9.21·16-s − 5.32·17-s − 8.56·18-s − 7.54·19-s + 2.01·20-s + 2.66·21-s + 6.68·22-s + 4.06·23-s + 18.0·24-s − 4.82·25-s − 9.63·26-s − 0.728·27-s − 5.07·28-s + ⋯ |
L(s) = 1 | − 1.83·2-s − 1.44·3-s + 2.38·4-s + 0.188·5-s + 2.66·6-s − 0.401·7-s − 2.54·8-s + 1.09·9-s − 0.347·10-s − 0.775·11-s − 3.45·12-s + 1.02·13-s + 0.739·14-s − 0.273·15-s + 2.30·16-s − 1.29·17-s − 2.01·18-s − 1.73·19-s + 0.450·20-s + 0.581·21-s + 1.42·22-s + 0.848·23-s + 3.69·24-s − 0.964·25-s − 1.88·26-s − 0.140·27-s − 0.958·28-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
\(L(1)\) |
\(\approx\) |
\(0.02194010780\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.02194010780\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.60T + 2T^{2} \) |
| 3 | \( 1 + 2.50T + 3T^{2} \) |
| 5 | \( 1 - 0.421T + 5T^{2} \) |
| 7 | \( 1 + 1.06T + 7T^{2} \) |
| 11 | \( 1 + 2.57T + 11T^{2} \) |
| 13 | \( 1 - 3.70T + 13T^{2} \) |
| 17 | \( 1 + 5.32T + 17T^{2} \) |
| 19 | \( 1 + 7.54T + 19T^{2} \) |
| 23 | \( 1 - 4.06T + 23T^{2} \) |
| 29 | \( 1 + 2.06T + 29T^{2} \) |
| 31 | \( 1 + 5.40T + 31T^{2} \) |
| 37 | \( 1 + 8.73T + 37T^{2} \) |
| 41 | \( 1 + 5.87T + 41T^{2} \) |
| 43 | \( 1 + 5.00T + 43T^{2} \) |
| 47 | \( 1 - 4.68T + 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 - 12.4T + 59T^{2} \) |
| 61 | \( 1 + 10.1T + 61T^{2} \) |
| 67 | \( 1 + 10.2T + 67T^{2} \) |
| 71 | \( 1 + 6.16T + 71T^{2} \) |
| 73 | \( 1 - 11.7T + 73T^{2} \) |
| 79 | \( 1 + 0.169T + 79T^{2} \) |
| 83 | \( 1 - 4.24T + 83T^{2} \) |
| 89 | \( 1 + 4.85T + 89T^{2} \) |
| 97 | \( 1 + 8.55T + 97T^{2} \) |
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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
−8.293889257207364327642554882244, −7.30812462655401317034243054323, −6.64854368461491995081946396747, −6.30096096745267201678012948129, −5.59314649069738458336434494446, −4.65677699493543536908212710493, −3.44522210740569745650712479825, −2.21322245080769783953834934685, −1.50949799980083696756294758298, −0.11074454048828744680802961144,
0.11074454048828744680802961144, 1.50949799980083696756294758298, 2.21322245080769783953834934685, 3.44522210740569745650712479825, 4.65677699493543536908212710493, 5.59314649069738458336434494446, 6.30096096745267201678012948129, 6.64854368461491995081946396747, 7.30812462655401317034243054323, 8.293889257207364327642554882244