| L(s) = 1 | − 0.650·2-s − 1.14·3-s − 1.57·4-s − 5-s + 0.742·6-s + 2.32·8-s − 1.69·9-s + 0.650·10-s + 11-s + 1.79·12-s − 6.40·13-s + 1.14·15-s + 1.63·16-s − 6.41·17-s + 1.10·18-s − 0.0447·19-s + 1.57·20-s − 0.650·22-s − 1.63·23-s − 2.65·24-s + 25-s + 4.16·26-s + 5.36·27-s − 1.63·29-s − 0.742·30-s + 1.00·31-s − 5.72·32-s + ⋯ |
| L(s) = 1 | − 0.460·2-s − 0.659·3-s − 0.788·4-s − 0.447·5-s + 0.303·6-s + 0.822·8-s − 0.565·9-s + 0.205·10-s + 0.301·11-s + 0.519·12-s − 1.77·13-s + 0.294·15-s + 0.409·16-s − 1.55·17-s + 0.260·18-s − 0.0102·19-s + 0.352·20-s − 0.138·22-s − 0.340·23-s − 0.542·24-s + 0.200·25-s + 0.817·26-s + 1.03·27-s − 0.304·29-s − 0.135·30-s + 0.179·31-s − 1.01·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1771511583\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1771511583\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 + 0.650T + 2T^{2} \) |
| 3 | \( 1 + 1.14T + 3T^{2} \) |
| 13 | \( 1 + 6.40T + 13T^{2} \) |
| 17 | \( 1 + 6.41T + 17T^{2} \) |
| 19 | \( 1 + 0.0447T + 19T^{2} \) |
| 23 | \( 1 + 1.63T + 23T^{2} \) |
| 29 | \( 1 + 1.63T + 29T^{2} \) |
| 31 | \( 1 - 1.00T + 31T^{2} \) |
| 37 | \( 1 + 3.17T + 37T^{2} \) |
| 41 | \( 1 + 11.8T + 41T^{2} \) |
| 43 | \( 1 + 10.4T + 43T^{2} \) |
| 47 | \( 1 - 9.84T + 47T^{2} \) |
| 53 | \( 1 + 6.57T + 53T^{2} \) |
| 59 | \( 1 - 2.05T + 59T^{2} \) |
| 61 | \( 1 + 10.7T + 61T^{2} \) |
| 67 | \( 1 - 10.1T + 67T^{2} \) |
| 71 | \( 1 - 8.91T + 71T^{2} \) |
| 73 | \( 1 + 11.6T + 73T^{2} \) |
| 79 | \( 1 + 4.67T + 79T^{2} \) |
| 83 | \( 1 + 15.9T + 83T^{2} \) |
| 89 | \( 1 + 9.24T + 89T^{2} \) |
| 97 | \( 1 + 9.80T + 97T^{2} \) |
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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
−8.722860579455578115102324360518, −8.323037365513571329020595430709, −7.28022972235074252024004144171, −6.73792153003236389674264259614, −5.61561948208500913499074193954, −4.84185606625276341273595537911, −4.37056796306065634967917043586, −3.16570189687984079558526283217, −1.92334764120090813960725039843, −0.28030472158565415966201422320,
0.28030472158565415966201422320, 1.92334764120090813960725039843, 3.16570189687984079558526283217, 4.37056796306065634967917043586, 4.84185606625276341273595537911, 5.61561948208500913499074193954, 6.73792153003236389674264259614, 7.28022972235074252024004144171, 8.323037365513571329020595430709, 8.722860579455578115102324360518
