| L(s) = 1 | + 3-s + 5-s + 1.48·7-s + 9-s + 1.67·11-s + 1.06·13-s + 15-s − 4.17·17-s − 1.78·19-s + 1.48·21-s − 0.687·23-s + 25-s + 27-s + 4.17·29-s − 0.0242·31-s + 1.67·33-s + 1.48·35-s + 3.83·37-s + 1.06·39-s + 10.0·41-s + 7.41·43-s + 45-s + 3.36·47-s − 4.78·49-s − 4.17·51-s + 9.22·53-s + 1.67·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.562·7-s + 0.333·9-s + 0.505·11-s + 0.294·13-s + 0.258·15-s − 1.01·17-s − 0.410·19-s + 0.324·21-s − 0.143·23-s + 0.200·25-s + 0.192·27-s + 0.775·29-s − 0.00436·31-s + 0.291·33-s + 0.251·35-s + 0.631·37-s + 0.170·39-s + 1.56·41-s + 1.13·43-s + 0.149·45-s + 0.491·47-s − 0.683·49-s − 0.584·51-s + 1.26·53-s + 0.226·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.979167485\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.979167485\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 67 | \( 1 + T \) |
| good | 7 | \( 1 - 1.48T + 7T^{2} \) |
| 11 | \( 1 - 1.67T + 11T^{2} \) |
| 13 | \( 1 - 1.06T + 13T^{2} \) |
| 17 | \( 1 + 4.17T + 17T^{2} \) |
| 19 | \( 1 + 1.78T + 19T^{2} \) |
| 23 | \( 1 + 0.687T + 23T^{2} \) |
| 29 | \( 1 - 4.17T + 29T^{2} \) |
| 31 | \( 1 + 0.0242T + 31T^{2} \) |
| 37 | \( 1 - 3.83T + 37T^{2} \) |
| 41 | \( 1 - 10.0T + 41T^{2} \) |
| 43 | \( 1 - 7.41T + 43T^{2} \) |
| 47 | \( 1 - 3.36T + 47T^{2} \) |
| 53 | \( 1 - 9.22T + 53T^{2} \) |
| 59 | \( 1 - 8.62T + 59T^{2} \) |
| 61 | \( 1 + 7.04T + 61T^{2} \) |
| 71 | \( 1 - 0.809T + 71T^{2} \) |
| 73 | \( 1 - 5.89T + 73T^{2} \) |
| 79 | \( 1 + 1.93T + 79T^{2} \) |
| 83 | \( 1 - 4.31T + 83T^{2} \) |
| 89 | \( 1 + 4.77T + 89T^{2} \) |
| 97 | \( 1 - 10.4T + 97T^{2} \) |
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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
−8.488002046105745081545396696192, −7.84303432808913008669249652103, −6.96599764376554954430732212428, −6.31090095430509610284679025510, −5.50879836590883268630642763467, −4.45602669073137979134822931207, −4.01366153742333132445091427138, −2.76080747749366082121384861344, −2.08236890162550031217679510828, −1.01233301282380661302842048901,
1.01233301282380661302842048901, 2.08236890162550031217679510828, 2.76080747749366082121384861344, 4.01366153742333132445091427138, 4.45602669073137979134822931207, 5.50879836590883268630642763467, 6.31090095430509610284679025510, 6.96599764376554954430732212428, 7.84303432808913008669249652103, 8.488002046105745081545396696192
