| L(s) = 1 | + 2-s + 4-s + 1.60·5-s + 2.89·7-s + 8-s + 1.60·10-s + 11-s + 4.89·13-s + 2.89·14-s + 16-s + 5.74·17-s − 19-s + 1.60·20-s + 22-s + 7.74·23-s − 2.43·25-s + 4.89·26-s + 2.89·28-s − 5.34·29-s − 7.43·31-s + 32-s + 5.74·34-s + 4.63·35-s − 7.49·37-s − 38-s + 1.60·40-s + 2.05·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.716·5-s + 1.09·7-s + 0.353·8-s + 0.506·10-s + 0.301·11-s + 1.35·13-s + 0.772·14-s + 0.250·16-s + 1.39·17-s − 0.229·19-s + 0.358·20-s + 0.213·22-s + 1.61·23-s − 0.487·25-s + 0.959·26-s + 0.546·28-s − 0.993·29-s − 1.33·31-s + 0.176·32-s + 0.985·34-s + 0.782·35-s − 1.23·37-s − 0.162·38-s + 0.253·40-s + 0.321·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3762 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3762 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.538509199\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.538509199\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 - 1.60T + 5T^{2} \) |
| 7 | \( 1 - 2.89T + 7T^{2} \) |
| 13 | \( 1 - 4.89T + 13T^{2} \) |
| 17 | \( 1 - 5.74T + 17T^{2} \) |
| 23 | \( 1 - 7.74T + 23T^{2} \) |
| 29 | \( 1 + 5.34T + 29T^{2} \) |
| 31 | \( 1 + 7.43T + 31T^{2} \) |
| 37 | \( 1 + 7.49T + 37T^{2} \) |
| 41 | \( 1 - 2.05T + 41T^{2} \) |
| 43 | \( 1 + 10.6T + 43T^{2} \) |
| 47 | \( 1 - 7.08T + 47T^{2} \) |
| 53 | \( 1 + 8.32T + 53T^{2} \) |
| 59 | \( 1 - 2.54T + 59T^{2} \) |
| 61 | \( 1 - 13.4T + 61T^{2} \) |
| 67 | \( 1 + 10.3T + 67T^{2} \) |
| 71 | \( 1 + 14.9T + 71T^{2} \) |
| 73 | \( 1 + 3.74T + 73T^{2} \) |
| 79 | \( 1 + 8.69T + 79T^{2} \) |
| 83 | \( 1 - 6.10T + 83T^{2} \) |
| 89 | \( 1 + 3.20T + 89T^{2} \) |
| 97 | \( 1 + 6.98T + 97T^{2} \) |
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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
−8.556450702955784331222785277976, −7.63162302825029442104499674946, −7.01331555380641836309007651757, −5.99468198560774501128534429134, −5.52616891795948409511824633647, −4.87477921070852655905321057952, −3.81638242429452264299338139202, −3.19403288764435951052283397951, −1.83438081712256438813287935059, −1.33253935861584840610752798828,
1.33253935861584840610752798828, 1.83438081712256438813287935059, 3.19403288764435951052283397951, 3.81638242429452264299338139202, 4.87477921070852655905321057952, 5.52616891795948409511824633647, 5.99468198560774501128534429134, 7.01331555380641836309007651757, 7.63162302825029442104499674946, 8.556450702955784331222785277976
