| L(s) = 1 | − 2-s + 4-s − 5-s − 7-s − 8-s + 10-s + 4·11-s − 13-s + 14-s + 16-s − 19-s − 20-s − 4·22-s + 4·23-s + 25-s + 26-s − 28-s − 5·29-s − 9·31-s − 32-s + 35-s − 4·37-s + 38-s + 40-s + 7·41-s + 4·44-s − 4·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.377·7-s − 0.353·8-s + 0.316·10-s + 1.20·11-s − 0.277·13-s + 0.267·14-s + 1/4·16-s − 0.229·19-s − 0.223·20-s − 0.852·22-s + 0.834·23-s + 1/5·25-s + 0.196·26-s − 0.188·28-s − 0.928·29-s − 1.61·31-s − 0.176·32-s + 0.169·35-s − 0.657·37-s + 0.162·38-s + 0.158·40-s + 1.09·41-s + 0.603·44-s − 0.589·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 166410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 166410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 43 | \( 1 \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 15 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 5 T + p T^{2} \) |
| 79 | \( 1 - 9 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.25263097871870, −13.00618165258792, −12.32652145101990, −12.16362613318749, −11.30393626040177, −11.18764259095289, −10.76710368135198, −9.987883028539399, −9.567060422358406, −9.170262681839553, −8.842966575567777, −8.164323831466851, −7.752165510176305, −7.086970761783243, −6.811920442059227, −6.382098666487255, −5.590365525063971, −5.224251659075079, −4.421562852810628, −3.759025044808928, −3.520466583190910, −2.758124782691458, −2.032203825011435, −1.481276894208237, −0.7310996763984153, 0,
0.7310996763984153, 1.481276894208237, 2.032203825011435, 2.758124782691458, 3.520466583190910, 3.759025044808928, 4.421562852810628, 5.224251659075079, 5.590365525063971, 6.382098666487255, 6.811920442059227, 7.086970761783243, 7.752165510176305, 8.164323831466851, 8.842966575567777, 9.170262681839553, 9.567060422358406, 9.987883028539399, 10.76710368135198, 11.18764259095289, 11.30393626040177, 12.16362613318749, 12.32652145101990, 13.00618165258792, 13.25263097871870
