| L(s) = 1 | − 2.79·2-s − 3-s + 5.81·4-s + 5-s + 2.79·6-s + 1.96·7-s − 10.6·8-s + 9-s − 2.79·10-s − 3.14·11-s − 5.81·12-s + 3.21·13-s − 5.50·14-s − 15-s + 18.1·16-s + 3.68·17-s − 2.79·18-s − 1.95·19-s + 5.81·20-s − 1.96·21-s + 8.78·22-s − 1.34·23-s + 10.6·24-s + 25-s − 8.98·26-s − 27-s + 11.4·28-s + ⋯ |
| L(s) = 1 | − 1.97·2-s − 0.577·3-s + 2.90·4-s + 0.447·5-s + 1.14·6-s + 0.744·7-s − 3.77·8-s + 0.333·9-s − 0.884·10-s − 0.947·11-s − 1.67·12-s + 0.891·13-s − 1.47·14-s − 0.258·15-s + 4.54·16-s + 0.893·17-s − 0.658·18-s − 0.447·19-s + 1.30·20-s − 0.429·21-s + 1.87·22-s − 0.281·23-s + 2.17·24-s + 0.200·25-s − 1.76·26-s − 0.192·27-s + 2.16·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 401 | \( 1 - T \) |
| good | 2 | \( 1 + 2.79T + 2T^{2} \) |
| 7 | \( 1 - 1.96T + 7T^{2} \) |
| 11 | \( 1 + 3.14T + 11T^{2} \) |
| 13 | \( 1 - 3.21T + 13T^{2} \) |
| 17 | \( 1 - 3.68T + 17T^{2} \) |
| 19 | \( 1 + 1.95T + 19T^{2} \) |
| 23 | \( 1 + 1.34T + 23T^{2} \) |
| 29 | \( 1 + 4.28T + 29T^{2} \) |
| 31 | \( 1 + 3.70T + 31T^{2} \) |
| 37 | \( 1 + 1.93T + 37T^{2} \) |
| 41 | \( 1 + 8.47T + 41T^{2} \) |
| 43 | \( 1 - 4.37T + 43T^{2} \) |
| 47 | \( 1 - 7.21T + 47T^{2} \) |
| 53 | \( 1 - 0.185T + 53T^{2} \) |
| 59 | \( 1 - 11.6T + 59T^{2} \) |
| 61 | \( 1 + 2.43T + 61T^{2} \) |
| 67 | \( 1 - 13.9T + 67T^{2} \) |
| 71 | \( 1 + 11.0T + 71T^{2} \) |
| 73 | \( 1 - 4.19T + 73T^{2} \) |
| 79 | \( 1 + 17.0T + 79T^{2} \) |
| 83 | \( 1 + 10.5T + 83T^{2} \) |
| 89 | \( 1 - 5.25T + 89T^{2} \) |
| 97 | \( 1 + 10.5T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.905915571725888930660457002648, −7.23180219071423107640480164403, −6.57820302871778135815022612794, −5.64586734047782195977734185191, −5.41702962819786295598107006026, −3.83221507497712618974176776749, −2.73778658647570992457486285149, −1.84367311335169347718045526940, −1.18863401519122915726709425163, 0,
1.18863401519122915726709425163, 1.84367311335169347718045526940, 2.73778658647570992457486285149, 3.83221507497712618974176776749, 5.41702962819786295598107006026, 5.64586734047782195977734185191, 6.57820302871778135815022612794, 7.23180219071423107640480164403, 7.905915571725888930660457002648
