L(s) = 1 | − 2-s + 4-s − 5-s − 8-s + 10-s − 4·11-s − 2·13-s + 16-s + 2·17-s + 19-s − 20-s + 4·22-s − 4·23-s + 25-s + 2·26-s − 6·29-s − 4·31-s − 32-s − 2·34-s − 6·37-s − 38-s + 40-s + 10·41-s − 4·43-s − 4·44-s + 4·46-s − 12·47-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s − 1.20·11-s − 0.554·13-s + 1/4·16-s + 0.485·17-s + 0.229·19-s − 0.223·20-s + 0.852·22-s − 0.834·23-s + 1/5·25-s + 0.392·26-s − 1.11·29-s − 0.718·31-s − 0.176·32-s − 0.342·34-s − 0.986·37-s − 0.162·38-s + 0.158·40-s + 1.56·41-s − 0.609·43-s − 0.603·44-s + 0.589·46-s − 1.75·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 83790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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
−14.51222066721497, −14.07287096837761, −13.35197052072748, −12.77093985223054, −12.52911584631833, −11.89279537977242, −11.32296112585882, −11.00138578606255, −10.34808934609644, −9.986577427941002, −9.457079747679905, −8.962029393788038, −8.219897852642171, −7.919700679432117, −7.404740721071735, −7.105975413273134, −6.176045122629032, −5.798532453280269, −5.057260922912950, −4.693292875567288, −3.709656773436253, −3.307067176560663, −2.571707892822194, −1.965258118860357, −1.259802285035381, 0, 0,
1.259802285035381, 1.965258118860357, 2.571707892822194, 3.307067176560663, 3.709656773436253, 4.693292875567288, 5.057260922912950, 5.798532453280269, 6.176045122629032, 7.105975413273134, 7.404740721071735, 7.919700679432117, 8.219897852642171, 8.962029393788038, 9.457079747679905, 9.986577427941002, 10.34808934609644, 11.00138578606255, 11.32296112585882, 11.89279537977242, 12.52911584631833, 12.77093985223054, 13.35197052072748, 14.07287096837761, 14.51222066721497