| L(s) = 1 | + 5-s + 4.73·7-s + 0.360·11-s + 5.26·13-s − 0.370·17-s − 4.60·19-s + 23-s + 25-s − 0.939·29-s + 9.66·31-s + 4.73·35-s + 3.26·37-s − 5.29·41-s + 1.25·47-s + 15.4·49-s − 10.9·53-s + 0.360·55-s + 9.66·59-s + 9.71·61-s + 5.26·65-s − 7.07·67-s − 11.3·71-s + 0.745·73-s + 1.70·77-s + 0.415·79-s + 9.26·83-s − 0.370·85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.79·7-s + 0.108·11-s + 1.45·13-s − 0.0899·17-s − 1.05·19-s + 0.208·23-s + 0.200·25-s − 0.174·29-s + 1.73·31-s + 0.800·35-s + 0.537·37-s − 0.827·41-s + 0.183·47-s + 2.20·49-s − 1.50·53-s + 0.0486·55-s + 1.25·59-s + 1.24·61-s + 0.652·65-s − 0.863·67-s − 1.34·71-s + 0.0872·73-s + 0.194·77-s + 0.0467·79-s + 1.01·83-s − 0.0402·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.343073600\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.343073600\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 - 4.73T + 7T^{2} \) |
| 11 | \( 1 - 0.360T + 11T^{2} \) |
| 13 | \( 1 - 5.26T + 13T^{2} \) |
| 17 | \( 1 + 0.370T + 17T^{2} \) |
| 19 | \( 1 + 4.60T + 19T^{2} \) |
| 29 | \( 1 + 0.939T + 29T^{2} \) |
| 31 | \( 1 - 9.66T + 31T^{2} \) |
| 37 | \( 1 - 3.26T + 37T^{2} \) |
| 41 | \( 1 + 5.29T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 1.25T + 47T^{2} \) |
| 53 | \( 1 + 10.9T + 53T^{2} \) |
| 59 | \( 1 - 9.66T + 59T^{2} \) |
| 61 | \( 1 - 9.71T + 61T^{2} \) |
| 67 | \( 1 + 7.07T + 67T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 73 | \( 1 - 0.745T + 73T^{2} \) |
| 79 | \( 1 - 0.415T + 79T^{2} \) |
| 83 | \( 1 - 9.26T + 83T^{2} \) |
| 89 | \( 1 + 12.6T + 89T^{2} \) |
| 97 | \( 1 - 14.0T + 97T^{2} \) |
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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
−8.025934477765864123626538632131, −7.12701054666033135049347327987, −6.30345242303854103628117809745, −5.81566870805209977326073590116, −4.87222448308005404242540730084, −4.45637650079522595400150967894, −3.58958670578191695976521264071, −2.46755196347086650262562836273, −1.68837990145725578811305610802, −0.990846893053214022973120212946,
0.990846893053214022973120212946, 1.68837990145725578811305610802, 2.46755196347086650262562836273, 3.58958670578191695976521264071, 4.45637650079522595400150967894, 4.87222448308005404242540730084, 5.81566870805209977326073590116, 6.30345242303854103628117809745, 7.12701054666033135049347327987, 8.025934477765864123626538632131
