| L(s) = 1 | − 0.490·2-s − 3-s − 1.75·4-s + 0.513·5-s + 0.490·6-s − 7-s + 1.84·8-s + 9-s − 0.252·10-s + 5.96·11-s + 1.75·12-s + 1.98·13-s + 0.490·14-s − 0.513·15-s + 2.61·16-s + 6.46·17-s − 0.490·18-s − 6.60·19-s − 0.903·20-s + 21-s − 2.92·22-s − 2.40·23-s − 1.84·24-s − 4.73·25-s − 0.973·26-s − 27-s + 1.75·28-s + ⋯ |
| L(s) = 1 | − 0.347·2-s − 0.577·3-s − 0.879·4-s + 0.229·5-s + 0.200·6-s − 0.377·7-s + 0.652·8-s + 0.333·9-s − 0.0797·10-s + 1.79·11-s + 0.507·12-s + 0.550·13-s + 0.131·14-s − 0.132·15-s + 0.652·16-s + 1.56·17-s − 0.115·18-s − 1.51·19-s − 0.201·20-s + 0.218·21-s − 0.624·22-s − 0.502·23-s − 0.376·24-s − 0.947·25-s − 0.190·26-s − 0.192·27-s + 0.332·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 + T \) |
| good | 2 | \( 1 + 0.490T + 2T^{2} \) |
| 5 | \( 1 - 0.513T + 5T^{2} \) |
| 11 | \( 1 - 5.96T + 11T^{2} \) |
| 13 | \( 1 - 1.98T + 13T^{2} \) |
| 17 | \( 1 - 6.46T + 17T^{2} \) |
| 19 | \( 1 + 6.60T + 19T^{2} \) |
| 23 | \( 1 + 2.40T + 23T^{2} \) |
| 29 | \( 1 + 2.90T + 29T^{2} \) |
| 31 | \( 1 - 4.89T + 31T^{2} \) |
| 37 | \( 1 + 11.1T + 37T^{2} \) |
| 41 | \( 1 + 3.57T + 41T^{2} \) |
| 43 | \( 1 - 10.8T + 43T^{2} \) |
| 47 | \( 1 + 10.4T + 47T^{2} \) |
| 53 | \( 1 + 7.77T + 53T^{2} \) |
| 59 | \( 1 + 8.77T + 59T^{2} \) |
| 61 | \( 1 + 0.398T + 61T^{2} \) |
| 67 | \( 1 - 5.98T + 67T^{2} \) |
| 71 | \( 1 + 11.1T + 71T^{2} \) |
| 73 | \( 1 - 9.50T + 73T^{2} \) |
| 79 | \( 1 - 6.32T + 79T^{2} \) |
| 83 | \( 1 + 0.606T + 83T^{2} \) |
| 89 | \( 1 - 9.02T + 89T^{2} \) |
| 97 | \( 1 - 9.02T + 97T^{2} \) |
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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
−7.61464673374349807529357221311, −6.53159501719751185624488969328, −6.23879635492166129969568584020, −5.47593544551518680986162098920, −4.62148162730929134656143807551, −3.85573367444498036700018635311, −3.47964046566855924211733256974, −1.81498832027324247687802844702, −1.16287965464867707745911174420, 0,
1.16287965464867707745911174420, 1.81498832027324247687802844702, 3.47964046566855924211733256974, 3.85573367444498036700018635311, 4.62148162730929134656143807551, 5.47593544551518680986162098920, 6.23879635492166129969568584020, 6.53159501719751185624488969328, 7.61464673374349807529357221311
