| L(s) = 1 | − 5·5-s + 16.5·7-s − 72.5·11-s − 59.8·13-s + 15.8·17-s − 136.·19-s + 163.·23-s + 25·25-s + 11.5·29-s − 41.0·31-s − 82.8·35-s − 242.·37-s + 57.8·41-s − 264.·43-s + 599.·47-s − 68.3·49-s + 592.·53-s + 362.·55-s + 288.·59-s + 825.·61-s + 299.·65-s + 810.·67-s − 966.·71-s + 802.·73-s − 1.20e3·77-s − 1.16e3·79-s + 502.·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.894·7-s − 1.98·11-s − 1.27·13-s + 0.226·17-s − 1.64·19-s + 1.47·23-s + 0.200·25-s + 0.0742·29-s − 0.237·31-s − 0.400·35-s − 1.07·37-s + 0.220·41-s − 0.938·43-s + 1.86·47-s − 0.199·49-s + 1.53·53-s + 0.889·55-s + 0.637·59-s + 1.73·61-s + 0.571·65-s + 1.47·67-s − 1.61·71-s + 1.28·73-s − 1.77·77-s − 1.65·79-s + 0.664·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.208153995\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.208153995\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 5T \) |
| good | 7 | \( 1 - 16.5T + 343T^{2} \) |
| 11 | \( 1 + 72.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 59.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 15.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 136.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 163.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 11.5T + 2.43e4T^{2} \) |
| 31 | \( 1 + 41.0T + 2.97e4T^{2} \) |
| 37 | \( 1 + 242.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 57.8T + 6.89e4T^{2} \) |
| 43 | \( 1 + 264.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 599.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 592.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 288.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 825.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 810.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 966.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 802.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.16e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 502.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 8.76T + 7.04e5T^{2} \) |
| 97 | \( 1 - 138.T + 9.12e5T^{2} \) |
| 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
−8.757855783355281454348882614192, −8.256153343227553007196270474336, −7.45904165432239253170438262316, −6.90180306135124587314395945550, −5.37476456040034213585661403509, −5.07422858976221788837546610709, −4.12223725090773435903947837833, −2.78192678605135610709575732668, −2.11066638263079081332516033241, −0.50247450830768774043569814457,
0.50247450830768774043569814457, 2.11066638263079081332516033241, 2.78192678605135610709575732668, 4.12223725090773435903947837833, 5.07422858976221788837546610709, 5.37476456040034213585661403509, 6.90180306135124587314395945550, 7.45904165432239253170438262316, 8.256153343227553007196270474336, 8.757855783355281454348882614192
