| L(s) = 1 | + 3.28·3-s − 1.93·7-s + 7.80·9-s + 5.62·11-s − 2.07·13-s + 3.42·17-s + 19-s − 6.35·21-s − 5.35·23-s + 15.7·27-s + 1.09·29-s + 3.09·31-s + 18.5·33-s + 3.28·37-s − 6.80·39-s − 11.6·41-s − 0.501·43-s − 12.6·47-s − 3.26·49-s + 11.2·51-s + 3.01·53-s + 3.28·57-s − 2.35·59-s + 7.62·61-s − 15.0·63-s + 12.2·67-s − 17.6·69-s + ⋯ |
| L(s) = 1 | + 1.89·3-s − 0.730·7-s + 2.60·9-s + 1.69·11-s − 0.574·13-s + 0.830·17-s + 0.229·19-s − 1.38·21-s − 1.11·23-s + 3.03·27-s + 0.202·29-s + 0.555·31-s + 3.22·33-s + 0.540·37-s − 1.08·39-s − 1.81·41-s − 0.0764·43-s − 1.84·47-s − 0.466·49-s + 1.57·51-s + 0.413·53-s + 0.435·57-s − 0.306·59-s + 0.976·61-s − 1.89·63-s + 1.49·67-s − 2.12·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.617005029\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.617005029\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 3.28T + 3T^{2} \) |
| 7 | \( 1 + 1.93T + 7T^{2} \) |
| 11 | \( 1 - 5.62T + 11T^{2} \) |
| 13 | \( 1 + 2.07T + 13T^{2} \) |
| 17 | \( 1 - 3.42T + 17T^{2} \) |
| 23 | \( 1 + 5.35T + 23T^{2} \) |
| 29 | \( 1 - 1.09T + 29T^{2} \) |
| 31 | \( 1 - 3.09T + 31T^{2} \) |
| 37 | \( 1 - 3.28T + 37T^{2} \) |
| 41 | \( 1 + 11.6T + 41T^{2} \) |
| 43 | \( 1 + 0.501T + 43T^{2} \) |
| 47 | \( 1 + 12.6T + 47T^{2} \) |
| 53 | \( 1 - 3.01T + 53T^{2} \) |
| 59 | \( 1 + 2.35T + 59T^{2} \) |
| 61 | \( 1 - 7.62T + 61T^{2} \) |
| 67 | \( 1 - 12.2T + 67T^{2} \) |
| 71 | \( 1 - 14.3T + 71T^{2} \) |
| 73 | \( 1 + 2.87T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 5.35T + 83T^{2} \) |
| 89 | \( 1 - 8.35T + 89T^{2} \) |
| 97 | \( 1 - 3.01T + 97T^{2} \) |
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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
−9.333289615233141188616662535514, −8.373860710128365146512028189701, −7.924619990615411977971366935903, −6.87701303704460933934273181062, −6.43933901159237273433225510714, −4.89441625447526179531833013217, −3.76938485448122477379246609713, −3.46653409816443336706769462798, −2.37422277111233868413714740535, −1.35757910354703740290716756116,
1.35757910354703740290716756116, 2.37422277111233868413714740535, 3.46653409816443336706769462798, 3.76938485448122477379246609713, 4.89441625447526179531833013217, 6.43933901159237273433225510714, 6.87701303704460933934273181062, 7.924619990615411977971366935903, 8.373860710128365146512028189701, 9.333289615233141188616662535514
