| L(s) = 1 | + 2.67·3-s + 4.67·7-s + 4.14·9-s − 0.672·11-s + 1.14·13-s − 3.52·17-s + 5.52·19-s + 12.4·21-s + 3.81·23-s + 3.05·27-s − 29-s − 1.52·31-s − 1.79·33-s − 7.16·37-s + 3.05·39-s + 2.85·41-s − 8.96·43-s + 6.67·47-s + 14.8·49-s − 9.43·51-s − 10.4·53-s + 14.7·57-s + 10.7·59-s − 14.4·61-s + 19.3·63-s + 7.81·67-s + 10.2·69-s + ⋯ |
| L(s) = 1 | + 1.54·3-s + 1.76·7-s + 1.38·9-s − 0.202·11-s + 0.317·13-s − 0.855·17-s + 1.26·19-s + 2.72·21-s + 0.795·23-s + 0.588·27-s − 0.185·29-s − 0.274·31-s − 0.313·33-s − 1.17·37-s + 0.489·39-s + 0.446·41-s − 1.36·43-s + 0.973·47-s + 2.11·49-s − 1.32·51-s − 1.44·53-s + 1.95·57-s + 1.40·59-s − 1.85·61-s + 2.43·63-s + 0.954·67-s + 1.22·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.083810876\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.083810876\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 - 2.67T + 3T^{2} \) |
| 7 | \( 1 - 4.67T + 7T^{2} \) |
| 11 | \( 1 + 0.672T + 11T^{2} \) |
| 13 | \( 1 - 1.14T + 13T^{2} \) |
| 17 | \( 1 + 3.52T + 17T^{2} \) |
| 19 | \( 1 - 5.52T + 19T^{2} \) |
| 23 | \( 1 - 3.81T + 23T^{2} \) |
| 31 | \( 1 + 1.52T + 31T^{2} \) |
| 37 | \( 1 + 7.16T + 37T^{2} \) |
| 41 | \( 1 - 2.85T + 41T^{2} \) |
| 43 | \( 1 + 8.96T + 43T^{2} \) |
| 47 | \( 1 - 6.67T + 47T^{2} \) |
| 53 | \( 1 + 10.4T + 53T^{2} \) |
| 59 | \( 1 - 10.7T + 59T^{2} \) |
| 61 | \( 1 + 14.4T + 61T^{2} \) |
| 67 | \( 1 - 7.81T + 67T^{2} \) |
| 71 | \( 1 - 4.48T + 71T^{2} \) |
| 73 | \( 1 - 4.96T + 73T^{2} \) |
| 79 | \( 1 - 2.38T + 79T^{2} \) |
| 83 | \( 1 + 14.0T + 83T^{2} \) |
| 89 | \( 1 + 1.63T + 89T^{2} \) |
| 97 | \( 1 - 9.32T + 97T^{2} \) |
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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
−8.738696014992058375394276501217, −8.054819222690493662782024551389, −7.57173710421289877330394848331, −6.82836326884977385490305499275, −5.43168642205040965272994990226, −4.80306705422168955343693419284, −3.90831595920615420844175131372, −3.03668636632188960428586115874, −2.09354704953540847285453925253, −1.34998284303655772911516223843,
1.34998284303655772911516223843, 2.09354704953540847285453925253, 3.03668636632188960428586115874, 3.90831595920615420844175131372, 4.80306705422168955343693419284, 5.43168642205040965272994990226, 6.82836326884977385490305499275, 7.57173710421289877330394848331, 8.054819222690493662782024551389, 8.738696014992058375394276501217
