| L(s) = 1 | + 2-s − 2.72·3-s + 4-s − 2.72·6-s − 4.14·7-s + 8-s + 4.45·9-s − 4.76·11-s − 2.72·12-s − 3.91·13-s − 4.14·14-s + 16-s + 3.31·17-s + 4.45·18-s − 1.85·19-s + 11.3·21-s − 4.76·22-s − 1.54·23-s − 2.72·24-s − 3.91·26-s − 3.96·27-s − 4.14·28-s + 8.87·29-s + 9.75·31-s + 32-s + 13.0·33-s + 3.31·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.57·3-s + 0.5·4-s − 1.11·6-s − 1.56·7-s + 0.353·8-s + 1.48·9-s − 1.43·11-s − 0.788·12-s − 1.08·13-s − 1.10·14-s + 0.250·16-s + 0.804·17-s + 1.04·18-s − 0.424·19-s + 2.46·21-s − 1.01·22-s − 0.321·23-s − 0.557·24-s − 0.768·26-s − 0.762·27-s − 0.783·28-s + 1.64·29-s + 1.75·31-s + 0.176·32-s + 2.26·33-s + 0.568·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7970439972\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7970439972\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 3 | \( 1 + 2.72T + 3T^{2} \) |
| 7 | \( 1 + 4.14T + 7T^{2} \) |
| 11 | \( 1 + 4.76T + 11T^{2} \) |
| 13 | \( 1 + 3.91T + 13T^{2} \) |
| 17 | \( 1 - 3.31T + 17T^{2} \) |
| 19 | \( 1 + 1.85T + 19T^{2} \) |
| 23 | \( 1 + 1.54T + 23T^{2} \) |
| 29 | \( 1 - 8.87T + 29T^{2} \) |
| 31 | \( 1 - 9.75T + 31T^{2} \) |
| 41 | \( 1 + 5.06T + 41T^{2} \) |
| 43 | \( 1 + 9.99T + 43T^{2} \) |
| 47 | \( 1 - 4.82T + 47T^{2} \) |
| 53 | \( 1 + 5.13T + 53T^{2} \) |
| 59 | \( 1 + 1.05T + 59T^{2} \) |
| 61 | \( 1 + 4.14T + 61T^{2} \) |
| 67 | \( 1 - 1.00T + 67T^{2} \) |
| 71 | \( 1 - 6.45T + 71T^{2} \) |
| 73 | \( 1 - 10.7T + 73T^{2} \) |
| 79 | \( 1 + 1.19T + 79T^{2} \) |
| 83 | \( 1 - 10.6T + 83T^{2} \) |
| 89 | \( 1 - 7.29T + 89T^{2} \) |
| 97 | \( 1 - 14.8T + 97T^{2} \) |
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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
−9.798829738217748380051029947283, −8.237634164788602682644172560848, −7.31485052323480244636394400067, −6.44382716403366498822880228439, −6.15252433879514551543924589270, −5.08319069364068668473291965189, −4.75367330997839420357848093515, −3.35127473752737026006467906167, −2.50959940832418229792012443377, −0.55536893386650606231007681147,
0.55536893386650606231007681147, 2.50959940832418229792012443377, 3.35127473752737026006467906167, 4.75367330997839420357848093515, 5.08319069364068668473291965189, 6.15252433879514551543924589270, 6.44382716403366498822880228439, 7.31485052323480244636394400067, 8.237634164788602682644172560848, 9.798829738217748380051029947283
