| L(s) = 1 | + 2-s + 0.140·3-s + 4-s + 0.140·6-s + 1.14·7-s + 8-s − 2.98·9-s + 3.12·11-s + 0.140·12-s + 2.85·13-s + 1.14·14-s + 16-s − 2.14·17-s − 2.98·18-s + 4.26·19-s + 0.160·21-s + 3.12·22-s + 1.71·23-s + 0.140·24-s + 2.85·26-s − 0.839·27-s + 1.14·28-s − 4.28·29-s + 6.40·31-s + 32-s + 0.438·33-s − 2.14·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.0810·3-s + 0.5·4-s + 0.0573·6-s + 0.431·7-s + 0.353·8-s − 0.993·9-s + 0.940·11-s + 0.0405·12-s + 0.793·13-s + 0.304·14-s + 0.250·16-s − 0.519·17-s − 0.702·18-s + 0.977·19-s + 0.0349·21-s + 0.665·22-s + 0.358·23-s + 0.0286·24-s + 0.560·26-s − 0.161·27-s + 0.215·28-s − 0.794·29-s + 1.14·31-s + 0.176·32-s + 0.0762·33-s − 0.367·34-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
| \(L(1)\) |
\(\approx\) |
\(3.074469805\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.074469805\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.140T + 3T^{2} \) |
| 7 | \( 1 - 1.14T + 7T^{2} \) |
| 11 | \( 1 - 3.12T + 11T^{2} \) |
| 13 | \( 1 - 2.85T + 13T^{2} \) |
| 17 | \( 1 + 2.14T + 17T^{2} \) |
| 19 | \( 1 - 4.26T + 19T^{2} \) |
| 23 | \( 1 - 1.71T + 23T^{2} \) |
| 29 | \( 1 + 4.28T + 29T^{2} \) |
| 31 | \( 1 - 6.40T + 31T^{2} \) |
| 41 | \( 1 + 3.24T + 41T^{2} \) |
| 43 | \( 1 - 10.6T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 7.96T + 53T^{2} \) |
| 59 | \( 1 - 2.69T + 59T^{2} \) |
| 61 | \( 1 - 4.57T + 61T^{2} \) |
| 67 | \( 1 - 10.3T + 67T^{2} \) |
| 71 | \( 1 + 7.10T + 71T^{2} \) |
| 73 | \( 1 + 0.261T + 73T^{2} \) |
| 79 | \( 1 + 8.52T + 79T^{2} \) |
| 83 | \( 1 + 1.16T + 83T^{2} \) |
| 89 | \( 1 + 13.6T + 89T^{2} \) |
| 97 | \( 1 - 14.2T + 97T^{2} \) |
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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
−9.085430276063378626463751307361, −8.512206978075076349174478737582, −7.61091526403152104785014718588, −6.70073776646501056286287484052, −5.95184061908039014224733690145, −5.23781168907357727327733294782, −4.22967748240561557188553702399, −3.43596770055792049299169169862, −2.44801223940381787647141426984, −1.16111311827070292245986629609,
1.16111311827070292245986629609, 2.44801223940381787647141426984, 3.43596770055792049299169169862, 4.22967748240561557188553702399, 5.23781168907357727327733294782, 5.95184061908039014224733690145, 6.70073776646501056286287484052, 7.61091526403152104785014718588, 8.512206978075076349174478737582, 9.085430276063378626463751307361
