| L(s) = 1 | − 2-s − 4-s + 3·5-s + 7-s + 3·8-s − 3·10-s + 7·13-s − 14-s − 16-s − 3·17-s − 2·19-s − 3·20-s + 4·23-s + 4·25-s − 7·26-s − 28-s − 7·29-s − 10·31-s − 5·32-s + 3·34-s + 3·35-s + 37-s + 2·38-s + 9·40-s + 5·41-s + 6·43-s − 4·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 1.34·5-s + 0.377·7-s + 1.06·8-s − 0.948·10-s + 1.94·13-s − 0.267·14-s − 1/4·16-s − 0.727·17-s − 0.458·19-s − 0.670·20-s + 0.834·23-s + 4/5·25-s − 1.37·26-s − 0.188·28-s − 1.29·29-s − 1.79·31-s − 0.883·32-s + 0.514·34-s + 0.507·35-s + 0.164·37-s + 0.324·38-s + 1.42·40-s + 0.780·41-s + 0.914·43-s − 0.589·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.840650925\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.840650925\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 7 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 + 19 T + p T^{2} \) |
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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
−8.092694130084279075817793160912, −7.23624080294519301798444468896, −6.51955891463457097547100099476, −5.62351731700509038739583933308, −5.39163292197067255609192841996, −4.20105370941441010036194450027, −3.70213581150642312299478685102, −2.30125816381599038226044777488, −1.65374019259496301387370869226, −0.813736600765810164694622109583,
0.813736600765810164694622109583, 1.65374019259496301387370869226, 2.30125816381599038226044777488, 3.70213581150642312299478685102, 4.20105370941441010036194450027, 5.39163292197067255609192841996, 5.62351731700509038739583933308, 6.51955891463457097547100099476, 7.23624080294519301798444468896, 8.092694130084279075817793160912
