| L(s) = 1 | − 0.618·2-s − 1.61·4-s − 1.61·7-s + 2.23·8-s + 0.763·11-s − 4.85·13-s + 1.00·14-s + 1.85·16-s + 0.763·17-s − 5.85·19-s − 0.472·22-s − 8.23·23-s + 3.00·26-s + 2.61·28-s + 1.38·29-s − 3·31-s − 5.61·32-s − 0.472·34-s + 4.23·37-s + 3.61·38-s + 5.23·41-s + 1.85·43-s − 1.23·44-s + 5.09·46-s + 1.61·47-s − 4.38·49-s + 7.85·52-s + ⋯ |
| L(s) = 1 | − 0.437·2-s − 0.809·4-s − 0.611·7-s + 0.790·8-s + 0.230·11-s − 1.34·13-s + 0.267·14-s + 0.463·16-s + 0.185·17-s − 1.34·19-s − 0.100·22-s − 1.71·23-s + 0.588·26-s + 0.494·28-s + 0.256·29-s − 0.538·31-s − 0.993·32-s − 0.0809·34-s + 0.696·37-s + 0.586·38-s + 0.817·41-s + 0.282·43-s − 0.186·44-s + 0.750·46-s + 0.236·47-s − 0.625·49-s + 1.08·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5049222013\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5049222013\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 0.618T + 2T^{2} \) |
| 7 | \( 1 + 1.61T + 7T^{2} \) |
| 11 | \( 1 - 0.763T + 11T^{2} \) |
| 13 | \( 1 + 4.85T + 13T^{2} \) |
| 17 | \( 1 - 0.763T + 17T^{2} \) |
| 19 | \( 1 + 5.85T + 19T^{2} \) |
| 23 | \( 1 + 8.23T + 23T^{2} \) |
| 29 | \( 1 - 1.38T + 29T^{2} \) |
| 31 | \( 1 + 3T + 31T^{2} \) |
| 37 | \( 1 - 4.23T + 37T^{2} \) |
| 41 | \( 1 - 5.23T + 41T^{2} \) |
| 43 | \( 1 - 1.85T + 43T^{2} \) |
| 47 | \( 1 - 1.61T + 47T^{2} \) |
| 53 | \( 1 + 5.47T + 53T^{2} \) |
| 59 | \( 1 - 4.14T + 59T^{2} \) |
| 61 | \( 1 + 4.70T + 61T^{2} \) |
| 67 | \( 1 - 9.23T + 67T^{2} \) |
| 71 | \( 1 - 4.38T + 71T^{2} \) |
| 73 | \( 1 + 9T + 73T^{2} \) |
| 79 | \( 1 - 3.09T + 79T^{2} \) |
| 83 | \( 1 - 1.76T + 83T^{2} \) |
| 89 | \( 1 + 8.94T + 89T^{2} \) |
| 97 | \( 1 - 2.85T + 97T^{2} \) |
| 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
−8.074349071074000633056813275290, −7.68080298067193976200345947968, −6.74400005210959597519863003510, −6.04311427947217401043014594141, −5.21369794580921075865830665697, −4.34892284328526705074840485293, −3.89494545184376969053271076590, −2.71990127901024809934324742470, −1.80725215332011272798032891092, −0.39566168884652893078027445784,
0.39566168884652893078027445784, 1.80725215332011272798032891092, 2.71990127901024809934324742470, 3.89494545184376969053271076590, 4.34892284328526705074840485293, 5.21369794580921075865830665697, 6.04311427947217401043014594141, 6.74400005210959597519863003510, 7.68080298067193976200345947968, 8.074349071074000633056813275290
