| L(s) = 1 | − 2-s + 4-s − 5-s + 7-s − 8-s + 10-s + 2·11-s + 13-s − 14-s + 16-s − 19-s − 20-s − 2·22-s + 23-s + 25-s − 26-s + 28-s − 32-s − 35-s − 2·37-s + 38-s + 40-s − 41-s + 2·44-s − 46-s + 47-s − 50-s + ⋯ |
| L(s) = 1 | − 2-s + 4-s − 5-s + 7-s − 8-s + 10-s + 2·11-s + 13-s − 14-s + 16-s − 19-s − 20-s − 2·22-s + 23-s + 25-s − 26-s + 28-s − 32-s − 35-s − 2·37-s + 38-s + 40-s − 41-s + 2·44-s − 46-s + 47-s − 50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8986013117\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8986013117\) |
| \(L(1)\) |
|
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 | \( 1 + T \) |
| good | 7 | \( 1 - T + T^{2} \) |
| 11 | \( ( 1 - T )^{2} \) |
| 13 | \( 1 - T + T^{2} \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( 1 + T + T^{2} \) |
| 23 | \( 1 - T + T^{2} \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 + T )^{2} \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( 1 - T + T^{2} \) |
| 53 | \( 1 - T + T^{2} \) |
| 59 | \( 1 + T + T^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 - T )^{2} \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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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
−8.716614620977279443224522644587, −8.386200373565016349787260905417, −7.39159207060500961261411305089, −6.81172104520950751945171665223, −6.16562420845847813048953437518, −4.95062801294588190106417223900, −3.96447592700757106755709514613, −3.36424030834846801203034182259, −1.85449462406173366536316977051, −1.07303712905675246062574686346,
1.07303712905675246062574686346, 1.85449462406173366536316977051, 3.36424030834846801203034182259, 3.96447592700757106755709514613, 4.95062801294588190106417223900, 6.16562420845847813048953437518, 6.81172104520950751945171665223, 7.39159207060500961261411305089, 8.386200373565016349787260905417, 8.716614620977279443224522644587
