| L(s) = 1 | − 0.149·2-s − 3.21·3-s − 1.97·4-s − 1.07·5-s + 0.479·6-s + 3.58·7-s + 0.593·8-s + 7.36·9-s + 0.160·10-s − 11-s + 6.36·12-s − 0.534·14-s + 3.46·15-s + 3.86·16-s − 5.78·17-s − 1.09·18-s + 7.32·19-s + 2.12·20-s − 11.5·21-s + 0.149·22-s − 5.23·23-s − 1.90·24-s − 3.84·25-s − 14.0·27-s − 7.08·28-s − 3.62·29-s − 0.516·30-s + ⋯ |
| L(s) = 1 | − 0.105·2-s − 1.85·3-s − 0.988·4-s − 0.481·5-s + 0.195·6-s + 1.35·7-s + 0.209·8-s + 2.45·9-s + 0.0507·10-s − 0.301·11-s + 1.83·12-s − 0.142·14-s + 0.894·15-s + 0.966·16-s − 1.40·17-s − 0.258·18-s + 1.68·19-s + 0.475·20-s − 2.51·21-s + 0.0317·22-s − 1.09·23-s − 0.389·24-s − 0.768·25-s − 2.70·27-s − 1.33·28-s − 0.673·29-s − 0.0942·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5174803833\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5174803833\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 0.149T + 2T^{2} \) |
| 3 | \( 1 + 3.21T + 3T^{2} \) |
| 5 | \( 1 + 1.07T + 5T^{2} \) |
| 7 | \( 1 - 3.58T + 7T^{2} \) |
| 17 | \( 1 + 5.78T + 17T^{2} \) |
| 19 | \( 1 - 7.32T + 19T^{2} \) |
| 23 | \( 1 + 5.23T + 23T^{2} \) |
| 29 | \( 1 + 3.62T + 29T^{2} \) |
| 31 | \( 1 - 2.65T + 31T^{2} \) |
| 37 | \( 1 - 3.91T + 37T^{2} \) |
| 41 | \( 1 - 1.04T + 41T^{2} \) |
| 43 | \( 1 + 1.21T + 43T^{2} \) |
| 47 | \( 1 + 11.9T + 47T^{2} \) |
| 53 | \( 1 - 0.145T + 53T^{2} \) |
| 59 | \( 1 - 2.35T + 59T^{2} \) |
| 61 | \( 1 - 9.46T + 61T^{2} \) |
| 67 | \( 1 - 5.14T + 67T^{2} \) |
| 71 | \( 1 - 0.0702T + 71T^{2} \) |
| 73 | \( 1 - 10.7T + 73T^{2} \) |
| 79 | \( 1 - 2.20T + 79T^{2} \) |
| 83 | \( 1 - 3.82T + 83T^{2} \) |
| 89 | \( 1 + 10.4T + 89T^{2} \) |
| 97 | \( 1 + 11.8T + 97T^{2} \) |
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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
−9.488139236300258524895323945743, −8.206628372796089582510960115248, −7.76686846394379344306974344771, −6.81188376745949010515405793844, −5.74245486609168263072597263089, −5.15735087195573803359818107100, −4.56038369286005811878891656327, −3.88070320560026231265100645573, −1.73560267627129356852371844168, −0.56725175972322881397529685349,
0.56725175972322881397529685349, 1.73560267627129356852371844168, 3.88070320560026231265100645573, 4.56038369286005811878891656327, 5.15735087195573803359818107100, 5.74245486609168263072597263089, 6.81188376745949010515405793844, 7.76686846394379344306974344771, 8.206628372796089582510960115248, 9.488139236300258524895323945743
