| L(s) = 1 | − 2.38·2-s − 0.344·3-s + 3.67·4-s + 0.273·5-s + 0.821·6-s + 0.727·7-s − 3.99·8-s − 2.88·9-s − 0.652·10-s + 3.88·11-s − 1.26·12-s − 3.75·13-s − 1.73·14-s − 0.0944·15-s + 2.16·16-s − 5.98·17-s + 6.86·18-s + 5.86·19-s + 1.00·20-s − 0.250·21-s − 9.25·22-s − 2.71·23-s + 1.37·24-s − 4.92·25-s + 8.94·26-s + 2.02·27-s + 2.67·28-s + ⋯ |
| L(s) = 1 | − 1.68·2-s − 0.199·3-s + 1.83·4-s + 0.122·5-s + 0.335·6-s + 0.274·7-s − 1.41·8-s − 0.960·9-s − 0.206·10-s + 1.17·11-s − 0.365·12-s − 1.04·13-s − 0.463·14-s − 0.0243·15-s + 0.541·16-s − 1.45·17-s + 1.61·18-s + 1.34·19-s + 0.225·20-s − 0.0547·21-s − 1.97·22-s − 0.566·23-s + 0.281·24-s − 0.985·25-s + 1.75·26-s + 0.390·27-s + 0.505·28-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
| \(L(1)\) |
\(\approx\) |
\(0.4499790651\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4499790651\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.38T + 2T^{2} \) |
| 3 | \( 1 + 0.344T + 3T^{2} \) |
| 5 | \( 1 - 0.273T + 5T^{2} \) |
| 7 | \( 1 - 0.727T + 7T^{2} \) |
| 11 | \( 1 - 3.88T + 11T^{2} \) |
| 13 | \( 1 + 3.75T + 13T^{2} \) |
| 17 | \( 1 + 5.98T + 17T^{2} \) |
| 19 | \( 1 - 5.86T + 19T^{2} \) |
| 23 | \( 1 + 2.71T + 23T^{2} \) |
| 29 | \( 1 + 1.73T + 29T^{2} \) |
| 31 | \( 1 + 10.6T + 31T^{2} \) |
| 37 | \( 1 + 0.0898T + 37T^{2} \) |
| 41 | \( 1 + 0.491T + 41T^{2} \) |
| 43 | \( 1 - 7.22T + 43T^{2} \) |
| 47 | \( 1 + 5.73T + 47T^{2} \) |
| 53 | \( 1 - 11.0T + 53T^{2} \) |
| 59 | \( 1 + 5.61T + 59T^{2} \) |
| 61 | \( 1 + 7.48T + 61T^{2} \) |
| 67 | \( 1 - 5.34T + 67T^{2} \) |
| 71 | \( 1 + 1.48T + 71T^{2} \) |
| 73 | \( 1 - 9.78T + 73T^{2} \) |
| 79 | \( 1 + 1.65T + 79T^{2} \) |
| 83 | \( 1 + 12.9T + 83T^{2} \) |
| 89 | \( 1 - 16.6T + 89T^{2} \) |
| 97 | \( 1 - 5.76T + 97T^{2} \) |
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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
−8.153099148804906673385733255859, −7.48886421796686945136539937562, −6.96139986638294081276493706186, −6.17567148873318567452434495908, −5.46233799135176755555080337816, −4.46455065805808981585277804540, −3.40221135874839282598574134858, −2.28771603000763563183734391377, −1.71290567991124602484131193425, −0.44951927719471929181217941842,
0.44951927719471929181217941842, 1.71290567991124602484131193425, 2.28771603000763563183734391377, 3.40221135874839282598574134858, 4.46455065805808981585277804540, 5.46233799135176755555080337816, 6.17567148873318567452434495908, 6.96139986638294081276493706186, 7.48886421796686945136539937562, 8.153099148804906673385733255859
