| L(s) = 1 | + 2.41·2-s + 3.82·4-s + 3.82·5-s + 4.41·8-s + 9.24·10-s + 3.82·11-s + 2.82·13-s + 2.99·16-s + 3.65·17-s − 19-s + 14.6·20-s + 9.24·22-s + 1.82·23-s + 9.65·25-s + 6.82·26-s + 4.82·29-s − 3.17·31-s − 1.58·32-s + 8.82·34-s + 1.17·37-s − 2.41·38-s + 16.8·40-s − 7.65·41-s − 12.6·43-s + 14.6·44-s + 4.41·46-s − 6.17·47-s + ⋯ |
| L(s) = 1 | + 1.70·2-s + 1.91·4-s + 1.71·5-s + 1.56·8-s + 2.92·10-s + 1.15·11-s + 0.784·13-s + 0.749·16-s + 0.886·17-s − 0.229·19-s + 3.27·20-s + 1.97·22-s + 0.381·23-s + 1.93·25-s + 1.33·26-s + 0.896·29-s − 0.569·31-s − 0.280·32-s + 1.51·34-s + 0.192·37-s − 0.391·38-s + 2.67·40-s − 1.19·41-s − 1.93·43-s + 2.20·44-s + 0.650·46-s − 0.900·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8379 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8379 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.765344463\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.765344463\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - 2.41T + 2T^{2} \) |
| 5 | \( 1 - 3.82T + 5T^{2} \) |
| 11 | \( 1 - 3.82T + 11T^{2} \) |
| 13 | \( 1 - 2.82T + 13T^{2} \) |
| 17 | \( 1 - 3.65T + 17T^{2} \) |
| 23 | \( 1 - 1.82T + 23T^{2} \) |
| 29 | \( 1 - 4.82T + 29T^{2} \) |
| 31 | \( 1 + 3.17T + 31T^{2} \) |
| 37 | \( 1 - 1.17T + 37T^{2} \) |
| 41 | \( 1 + 7.65T + 41T^{2} \) |
| 43 | \( 1 + 12.6T + 43T^{2} \) |
| 47 | \( 1 + 6.17T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 + 14.6T + 61T^{2} \) |
| 67 | \( 1 + 12T + 67T^{2} \) |
| 71 | \( 1 + 4.34T + 71T^{2} \) |
| 73 | \( 1 + 10.3T + 73T^{2} \) |
| 79 | \( 1 - 2.34T + 79T^{2} \) |
| 83 | \( 1 + 4.17T + 83T^{2} \) |
| 89 | \( 1 + 10.8T + 89T^{2} \) |
| 97 | \( 1 - 16.8T + 97T^{2} \) |
| 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
−7.38339003687213147439767350464, −6.61456317085116609378301022167, −6.18534755255517844545005576251, −5.80122191701016133853148485879, −4.97447331974842749874321414018, −4.47582617202134937864740921630, −3.33567118835517075670017424799, −3.06424817070466360044516943899, −1.78494248570683923105349552154, −1.46228871865551032818498935455,
1.46228871865551032818498935455, 1.78494248570683923105349552154, 3.06424817070466360044516943899, 3.33567118835517075670017424799, 4.47582617202134937864740921630, 4.97447331974842749874321414018, 5.80122191701016133853148485879, 6.18534755255517844545005576251, 6.61456317085116609378301022167, 7.38339003687213147439767350464
