| L(s) = 1 | + 2-s + 4-s + 3·5-s + 7-s + 8-s + 3·10-s + 3·11-s + 13-s + 14-s + 16-s − 3·17-s + 5·19-s + 3·20-s + 3·22-s − 3·23-s + 4·25-s + 26-s + 28-s − 3·29-s − 10·31-s + 32-s − 3·34-s + 3·35-s − 7·37-s + 5·38-s + 3·40-s + 6·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.34·5-s + 0.377·7-s + 0.353·8-s + 0.948·10-s + 0.904·11-s + 0.277·13-s + 0.267·14-s + 1/4·16-s − 0.727·17-s + 1.14·19-s + 0.670·20-s + 0.639·22-s − 0.625·23-s + 4/5·25-s + 0.196·26-s + 0.188·28-s − 0.557·29-s − 1.79·31-s + 0.176·32-s − 0.514·34-s + 0.507·35-s − 1.15·37-s + 0.811·38-s + 0.474·40-s + 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.727883447\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.727883447\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−9.324599393283520040395095380365, −8.841848699193986448317936408324, −7.53876669465829471955650703578, −6.83898930928988508530993461842, −5.85174835642101621758697196256, −5.52063493483957242179120678609, −4.39709509400492818170140602730, −3.48277098992130039606057870827, −2.22618758276738166969303174205, −1.46015232310843674903691285794,
1.46015232310843674903691285794, 2.22618758276738166969303174205, 3.48277098992130039606057870827, 4.39709509400492818170140602730, 5.52063493483957242179120678609, 5.85174835642101621758697196256, 6.83898930928988508530993461842, 7.53876669465829471955650703578, 8.841848699193986448317936408324, 9.324599393283520040395095380365
