L(s) = 1 | − 0.603·2-s − 0.154·3-s − 1.63·4-s + 4.17·5-s + 0.0932·6-s + 2.73·7-s + 2.19·8-s − 2.97·9-s − 2.51·10-s + 11-s + 0.252·12-s − 1.65·14-s − 0.645·15-s + 1.94·16-s − 2.78·17-s + 1.79·18-s + 8.62·19-s − 6.83·20-s − 0.423·21-s − 0.603·22-s − 5.00·23-s − 0.338·24-s + 12.4·25-s + 0.923·27-s − 4.48·28-s + 5.70·29-s + 0.389·30-s + ⋯ |
L(s) = 1 | − 0.426·2-s − 0.0892·3-s − 0.818·4-s + 1.86·5-s + 0.0380·6-s + 1.03·7-s + 0.775·8-s − 0.992·9-s − 0.796·10-s + 0.301·11-s + 0.0729·12-s − 0.441·14-s − 0.166·15-s + 0.487·16-s − 0.676·17-s + 0.423·18-s + 1.97·19-s − 1.52·20-s − 0.0923·21-s − 0.128·22-s − 1.04·23-s − 0.0691·24-s + 2.48·25-s + 0.177·27-s − 0.847·28-s + 1.05·29-s + 0.0710·30-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
\(L(1)\) |
\(\approx\) |
\(1.778871171\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.778871171\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.603T + 2T^{2} \) |
| 3 | \( 1 + 0.154T + 3T^{2} \) |
| 5 | \( 1 - 4.17T + 5T^{2} \) |
| 7 | \( 1 - 2.73T + 7T^{2} \) |
| 17 | \( 1 + 2.78T + 17T^{2} \) |
| 19 | \( 1 - 8.62T + 19T^{2} \) |
| 23 | \( 1 + 5.00T + 23T^{2} \) |
| 29 | \( 1 - 5.70T + 29T^{2} \) |
| 31 | \( 1 - 6.49T + 31T^{2} \) |
| 37 | \( 1 + 3.15T + 37T^{2} \) |
| 41 | \( 1 - 1.15T + 41T^{2} \) |
| 43 | \( 1 + 6.42T + 43T^{2} \) |
| 47 | \( 1 + 0.354T + 47T^{2} \) |
| 53 | \( 1 + 11.4T + 53T^{2} \) |
| 59 | \( 1 + 3.55T + 59T^{2} \) |
| 61 | \( 1 - 9.99T + 61T^{2} \) |
| 67 | \( 1 - 8.44T + 67T^{2} \) |
| 71 | \( 1 - 5.99T + 71T^{2} \) |
| 73 | \( 1 + 4.94T + 73T^{2} \) |
| 79 | \( 1 + 7.84T + 79T^{2} \) |
| 83 | \( 1 - 2.14T + 83T^{2} \) |
| 89 | \( 1 - 6.72T + 89T^{2} \) |
| 97 | \( 1 + 10.5T + 97T^{2} \) |
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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
−9.351572875605977727119933034860, −8.508397298135571769748385050847, −8.042371565190255061582981206963, −6.76557787990542458277437127908, −5.88917315005032874348923095286, −5.19971286199733425466784230653, −4.67122193215628608733228533171, −3.13379495702853415176306315620, −1.98581721617233015786765043763, −1.06125610377382545421221103059,
1.06125610377382545421221103059, 1.98581721617233015786765043763, 3.13379495702853415176306315620, 4.67122193215628608733228533171, 5.19971286199733425466784230653, 5.88917315005032874348923095286, 6.76557787990542458277437127908, 8.042371565190255061582981206963, 8.508397298135571769748385050847, 9.351572875605977727119933034860