| L(s) = 1 | − 3.01·3-s + 5-s + 7-s + 6.08·9-s + 0.174·11-s − 13-s − 3.01·15-s + 3.58·17-s + 6.28·19-s − 3.01·21-s − 4.20·23-s + 25-s − 9.29·27-s + 9.21·29-s − 1.01·31-s − 0.526·33-s + 35-s + 2.31·37-s + 3.01·39-s + 0.943·41-s − 2.46·43-s + 6.08·45-s − 5.15·47-s + 49-s − 10.8·51-s + 1.62·53-s + 0.174·55-s + ⋯ |
| L(s) = 1 | − 1.74·3-s + 0.447·5-s + 0.377·7-s + 2.02·9-s + 0.0526·11-s − 0.277·13-s − 0.778·15-s + 0.869·17-s + 1.44·19-s − 0.657·21-s − 0.876·23-s + 0.200·25-s − 1.78·27-s + 1.71·29-s − 0.182·31-s − 0.0916·33-s + 0.169·35-s + 0.380·37-s + 0.482·39-s + 0.147·41-s − 0.375·43-s + 0.907·45-s − 0.751·47-s + 0.142·49-s − 1.51·51-s + 0.222·53-s + 0.0235·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.216922321\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.216922321\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + 3.01T + 3T^{2} \) |
| 11 | \( 1 - 0.174T + 11T^{2} \) |
| 17 | \( 1 - 3.58T + 17T^{2} \) |
| 19 | \( 1 - 6.28T + 19T^{2} \) |
| 23 | \( 1 + 4.20T + 23T^{2} \) |
| 29 | \( 1 - 9.21T + 29T^{2} \) |
| 31 | \( 1 + 1.01T + 31T^{2} \) |
| 37 | \( 1 - 2.31T + 37T^{2} \) |
| 41 | \( 1 - 0.943T + 41T^{2} \) |
| 43 | \( 1 + 2.46T + 43T^{2} \) |
| 47 | \( 1 + 5.15T + 47T^{2} \) |
| 53 | \( 1 - 1.62T + 53T^{2} \) |
| 59 | \( 1 - 2.13T + 59T^{2} \) |
| 61 | \( 1 + 5.53T + 61T^{2} \) |
| 67 | \( 1 - 9.87T + 67T^{2} \) |
| 71 | \( 1 + 3.48T + 71T^{2} \) |
| 73 | \( 1 - 2.36T + 73T^{2} \) |
| 79 | \( 1 + 8.73T + 79T^{2} \) |
| 83 | \( 1 + 9.38T + 83T^{2} \) |
| 89 | \( 1 + 5.33T + 89T^{2} \) |
| 97 | \( 1 + 18.2T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.440447448338688159877670733269, −7.60570813628639160974760966365, −6.90289371154764911067173200440, −6.15284610843159423666861467798, −5.52854408907403521248584898965, −5.00335474955079657296514860295, −4.23051630208865556134718731114, −3.00511980139458752690528065390, −1.61205095963830497637144204505, −0.75607962694695306304456230738,
0.75607962694695306304456230738, 1.61205095963830497637144204505, 3.00511980139458752690528065390, 4.23051630208865556134718731114, 5.00335474955079657296514860295, 5.52854408907403521248584898965, 6.15284610843159423666861467798, 6.90289371154764911067173200440, 7.60570813628639160974760966365, 8.440447448338688159877670733269
