| L(s) = 1 | − 2.74·2-s + 5.52·4-s + 1.10·7-s − 9.68·8-s − 0.0961·11-s − 1.00·13-s − 3.03·14-s + 15.5·16-s − 1.92·17-s + 1.36·19-s + 0.263·22-s − 1.36·23-s + 2.76·26-s + 6.10·28-s + 29-s + 2.17·31-s − 23.1·32-s + 5.28·34-s − 6.61·37-s − 3.75·38-s − 5.07·41-s + 7.53·43-s − 0.531·44-s + 3.74·46-s − 5.77·47-s − 5.78·49-s − 5.57·52-s + ⋯ |
| L(s) = 1 | − 1.94·2-s + 2.76·4-s + 0.417·7-s − 3.42·8-s − 0.0289·11-s − 0.279·13-s − 0.809·14-s + 3.87·16-s − 0.467·17-s + 0.314·19-s + 0.0562·22-s − 0.284·23-s + 0.542·26-s + 1.15·28-s + 0.185·29-s + 0.390·31-s − 4.09·32-s + 0.906·34-s − 1.08·37-s − 0.609·38-s − 0.793·41-s + 1.14·43-s − 0.0801·44-s + 0.552·46-s − 0.843·47-s − 0.825·49-s − 0.772·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + 2.74T + 2T^{2} \) |
| 7 | \( 1 - 1.10T + 7T^{2} \) |
| 11 | \( 1 + 0.0961T + 11T^{2} \) |
| 13 | \( 1 + 1.00T + 13T^{2} \) |
| 17 | \( 1 + 1.92T + 17T^{2} \) |
| 19 | \( 1 - 1.36T + 19T^{2} \) |
| 23 | \( 1 + 1.36T + 23T^{2} \) |
| 31 | \( 1 - 2.17T + 31T^{2} \) |
| 37 | \( 1 + 6.61T + 37T^{2} \) |
| 41 | \( 1 + 5.07T + 41T^{2} \) |
| 43 | \( 1 - 7.53T + 43T^{2} \) |
| 47 | \( 1 + 5.77T + 47T^{2} \) |
| 53 | \( 1 - 2.54T + 53T^{2} \) |
| 59 | \( 1 - 12.6T + 59T^{2} \) |
| 61 | \( 1 - 7.29T + 61T^{2} \) |
| 67 | \( 1 - 2.77T + 67T^{2} \) |
| 71 | \( 1 - 6.05T + 71T^{2} \) |
| 73 | \( 1 + 11.5T + 73T^{2} \) |
| 79 | \( 1 + 3.01T + 79T^{2} \) |
| 83 | \( 1 - 0.455T + 83T^{2} \) |
| 89 | \( 1 + 7.57T + 89T^{2} \) |
| 97 | \( 1 - 10.7T + 97T^{2} \) |
| 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
−7.84483696562995772229409665275, −7.13451706664709801126025684469, −6.66578940633069557172623322128, −5.82174389419070202400229899355, −4.97914938901750851800590969850, −3.72506426892479960409363037273, −2.72527112973358278385329256965, −1.99917436100928333396203774909, −1.12823699302895579248435988605, 0,
1.12823699302895579248435988605, 1.99917436100928333396203774909, 2.72527112973358278385329256965, 3.72506426892479960409363037273, 4.97914938901750851800590969850, 5.82174389419070202400229899355, 6.66578940633069557172623322128, 7.13451706664709801126025684469, 7.84483696562995772229409665275
