| L(s) = 1 | − 2-s + 4-s + 2.56·5-s + 0.518·7-s − 8-s − 2.56·10-s − 11-s + 5.15·13-s − 0.518·14-s + 16-s − 1.91·17-s + 19-s + 2.56·20-s + 22-s + 3.15·23-s + 1.58·25-s − 5.15·26-s + 0.518·28-s + 5.72·29-s + 4.13·31-s − 32-s + 1.91·34-s + 1.32·35-s − 3.22·37-s − 38-s − 2.56·40-s − 6.76·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.14·5-s + 0.195·7-s − 0.353·8-s − 0.811·10-s − 0.301·11-s + 1.42·13-s − 0.138·14-s + 0.250·16-s − 0.465·17-s + 0.229·19-s + 0.573·20-s + 0.213·22-s + 0.657·23-s + 0.317·25-s − 1.01·26-s + 0.0978·28-s + 1.06·29-s + 0.741·31-s − 0.176·32-s + 0.328·34-s + 0.224·35-s − 0.530·37-s − 0.162·38-s − 0.405·40-s − 1.05·41-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
| \(L(1)\) |
\(\approx\) |
\(1.922391901\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.922391901\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.56T + 5T^{2} \) |
| 7 | \( 1 - 0.518T + 7T^{2} \) |
| 13 | \( 1 - 5.15T + 13T^{2} \) |
| 17 | \( 1 + 1.91T + 17T^{2} \) |
| 23 | \( 1 - 3.15T + 23T^{2} \) |
| 29 | \( 1 - 5.72T + 29T^{2} \) |
| 31 | \( 1 - 4.13T + 31T^{2} \) |
| 37 | \( 1 + 3.22T + 37T^{2} \) |
| 41 | \( 1 + 6.76T + 41T^{2} \) |
| 43 | \( 1 - 1.41T + 43T^{2} \) |
| 47 | \( 1 - 3.15T + 47T^{2} \) |
| 53 | \( 1 + 5.28T + 53T^{2} \) |
| 59 | \( 1 - 3.80T + 59T^{2} \) |
| 61 | \( 1 - 9.66T + 61T^{2} \) |
| 67 | \( 1 - 9.05T + 67T^{2} \) |
| 71 | \( 1 - 1.55T + 71T^{2} \) |
| 73 | \( 1 - 16.8T + 73T^{2} \) |
| 79 | \( 1 - 13.3T + 79T^{2} \) |
| 83 | \( 1 + 6.76T + 83T^{2} \) |
| 89 | \( 1 + 3.24T + 89T^{2} \) |
| 97 | \( 1 + 0.627T + 97T^{2} \) |
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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
−8.401853886414960820938873080650, −8.149495258236670254414009550864, −6.80707571614980637920830943503, −6.51234013298396827708508518230, −5.62206137955120138910412635931, −4.94451586660623075256407985526, −3.71778143240446719559358112572, −2.73078411361297741009895183957, −1.83458427168568337588595551000, −0.946547347695779516514762008210,
0.946547347695779516514762008210, 1.83458427168568337588595551000, 2.73078411361297741009895183957, 3.71778143240446719559358112572, 4.94451586660623075256407985526, 5.62206137955120138910412635931, 6.51234013298396827708508518230, 6.80707571614980637920830943503, 8.149495258236670254414009550864, 8.401853886414960820938873080650
