| L(s) = 1 | − 5-s + 3·7-s + 2.87·11-s + 4.87·13-s + 3.87·17-s + 4.87·19-s + 23-s + 25-s − 1.87·29-s + 3·31-s − 3·35-s + 37-s − 1.87·41-s − 11.7·43-s − 0.872·47-s + 2·49-s − 3.87·53-s − 2.87·55-s + 1.87·59-s + 1.12·61-s − 4.87·65-s − 4.74·67-s − 9.61·71-s + 4.87·73-s + 8.61·77-s + 4·79-s + 7.87·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.13·7-s + 0.866·11-s + 1.35·13-s + 0.939·17-s + 1.11·19-s + 0.208·23-s + 0.200·25-s − 0.347·29-s + 0.538·31-s − 0.507·35-s + 0.164·37-s − 0.292·41-s − 1.79·43-s − 0.127·47-s + 0.285·49-s − 0.531·53-s − 0.387·55-s + 0.243·59-s + 0.144·61-s − 0.604·65-s − 0.579·67-s − 1.14·71-s + 0.570·73-s + 0.982·77-s + 0.450·79-s + 0.864·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.559929358\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.559929358\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3T + 7T^{2} \) |
| 11 | \( 1 - 2.87T + 11T^{2} \) |
| 13 | \( 1 - 4.87T + 13T^{2} \) |
| 17 | \( 1 - 3.87T + 17T^{2} \) |
| 19 | \( 1 - 4.87T + 19T^{2} \) |
| 29 | \( 1 + 1.87T + 29T^{2} \) |
| 31 | \( 1 - 3T + 31T^{2} \) |
| 37 | \( 1 - T + 37T^{2} \) |
| 41 | \( 1 + 1.87T + 41T^{2} \) |
| 43 | \( 1 + 11.7T + 43T^{2} \) |
| 47 | \( 1 + 0.872T + 47T^{2} \) |
| 53 | \( 1 + 3.87T + 53T^{2} \) |
| 59 | \( 1 - 1.87T + 59T^{2} \) |
| 61 | \( 1 - 1.12T + 61T^{2} \) |
| 67 | \( 1 + 4.74T + 67T^{2} \) |
| 71 | \( 1 + 9.61T + 71T^{2} \) |
| 73 | \( 1 - 4.87T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 7.87T + 83T^{2} \) |
| 89 | \( 1 - 13.7T + 89T^{2} \) |
| 97 | \( 1 - 8T + 97T^{2} \) |
| 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
−8.257915665569655269676386070751, −7.88430841568634798803829326183, −7.00888863311170284146490419710, −6.22456825359184981668352813852, −5.37630802114382543692091451618, −4.68296146508467901807861057523, −3.75205009172150650063127011724, −3.18132630340438396823424824375, −1.66805280637585881445171681880, −1.03658110433159790286683807509,
1.03658110433159790286683807509, 1.66805280637585881445171681880, 3.18132630340438396823424824375, 3.75205009172150650063127011724, 4.68296146508467901807861057523, 5.37630802114382543692091451618, 6.22456825359184981668352813852, 7.00888863311170284146490419710, 7.88430841568634798803829326183, 8.257915665569655269676386070751
