| L(s) = 1 | + 2.63·2-s − 3-s + 4.95·4-s − 5-s − 2.63·6-s + 5.03·7-s + 7.78·8-s + 9-s − 2.63·10-s + 2.09·11-s − 4.95·12-s + 13.2·14-s + 15-s + 10.6·16-s + 3.94·17-s + 2.63·18-s − 1.13·19-s − 4.95·20-s − 5.03·21-s + 5.53·22-s − 9.28·23-s − 7.78·24-s + 25-s − 27-s + 24.9·28-s − 9.15·29-s + 2.63·30-s + ⋯ |
| L(s) = 1 | + 1.86·2-s − 0.577·3-s + 2.47·4-s − 0.447·5-s − 1.07·6-s + 1.90·7-s + 2.75·8-s + 0.333·9-s − 0.833·10-s + 0.632·11-s − 1.42·12-s + 3.54·14-s + 0.258·15-s + 2.65·16-s + 0.957·17-s + 0.621·18-s − 0.261·19-s − 1.10·20-s − 1.09·21-s + 1.17·22-s − 1.93·23-s − 1.58·24-s + 0.200·25-s − 0.192·27-s + 4.71·28-s − 1.69·29-s + 0.481·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.926911069\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.926911069\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 2.63T + 2T^{2} \) |
| 7 | \( 1 - 5.03T + 7T^{2} \) |
| 11 | \( 1 - 2.09T + 11T^{2} \) |
| 17 | \( 1 - 3.94T + 17T^{2} \) |
| 19 | \( 1 + 1.13T + 19T^{2} \) |
| 23 | \( 1 + 9.28T + 23T^{2} \) |
| 29 | \( 1 + 9.15T + 29T^{2} \) |
| 31 | \( 1 - 1.79T + 31T^{2} \) |
| 37 | \( 1 + 2.75T + 37T^{2} \) |
| 41 | \( 1 + 2.34T + 41T^{2} \) |
| 43 | \( 1 + 0.995T + 43T^{2} \) |
| 47 | \( 1 - 7.60T + 47T^{2} \) |
| 53 | \( 1 + 1.07T + 53T^{2} \) |
| 59 | \( 1 - 4.06T + 59T^{2} \) |
| 61 | \( 1 - 5.60T + 61T^{2} \) |
| 67 | \( 1 - 4.39T + 67T^{2} \) |
| 71 | \( 1 - 1.65T + 71T^{2} \) |
| 73 | \( 1 + 10.4T + 73T^{2} \) |
| 79 | \( 1 + 7.78T + 79T^{2} \) |
| 83 | \( 1 + 7.35T + 83T^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 - 17.4T + 97T^{2} \) |
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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
−8.601875385498627564116290602588, −7.70468721962683727641062606762, −7.31330709673517724686036276714, −6.20465814419401193275132309645, −5.55936276511623888135298039157, −4.97381990287534515648257800652, −4.12366560712757153163774376440, −3.74527554032167489137492850564, −2.21570456049888208092869774220, −1.44904745664812244938739464654,
1.44904745664812244938739464654, 2.21570456049888208092869774220, 3.74527554032167489137492850564, 4.12366560712757153163774376440, 4.97381990287534515648257800652, 5.55936276511623888135298039157, 6.20465814419401193275132309645, 7.31330709673517724686036276714, 7.70468721962683727641062606762, 8.601875385498627564116290602588
