| L(s) = 1 | − 5·5-s + 7.61·7-s + 12.5·11-s − 20.3·13-s − 22.6·17-s − 18.8·19-s − 6.29·23-s + 25·25-s + 206.·29-s + 33.7·31-s − 38.0·35-s + 13.4·37-s − 290.·41-s − 338.·43-s + 263.·47-s − 285.·49-s + 240.·53-s − 62.8·55-s − 373.·59-s − 454.·61-s + 101.·65-s + 442.·67-s + 371.·71-s + 901.·73-s + 95.7·77-s − 1.03e3·79-s + 34.6·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.410·7-s + 0.344·11-s − 0.434·13-s − 0.323·17-s − 0.227·19-s − 0.0570·23-s + 0.200·25-s + 1.32·29-s + 0.195·31-s − 0.183·35-s + 0.0598·37-s − 1.10·41-s − 1.20·43-s + 0.816·47-s − 0.831·49-s + 0.622·53-s − 0.154·55-s − 0.824·59-s − 0.954·61-s + 0.194·65-s + 0.807·67-s + 0.620·71-s + 1.44·73-s + 0.141·77-s − 1.47·79-s + 0.0458·83-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
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 7.61T + 343T^{2} \) |
| 11 | \( 1 - 12.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 20.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 22.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 18.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 6.29T + 1.21e4T^{2} \) |
| 29 | \( 1 - 206.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 33.7T + 2.97e4T^{2} \) |
| 37 | \( 1 - 13.4T + 5.06e4T^{2} \) |
| 41 | \( 1 + 290.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 338.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 263.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 240.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 373.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 454.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 442.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 371.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 901.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.03e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 34.6T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.07e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.38e3T + 9.12e5T^{2} \) |
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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
−8.502339747541824544278461919128, −7.970922414955782464239708014317, −6.98396464687216219598665428081, −6.36010954554772520280589166210, −5.16193887071093543259118065962, −4.50817584194566389541083672301, −3.53789753929419943789888133059, −2.46558670167412095469957373115, −1.29097733346499617200326729960, 0,
1.29097733346499617200326729960, 2.46558670167412095469957373115, 3.53789753929419943789888133059, 4.50817584194566389541083672301, 5.16193887071093543259118065962, 6.36010954554772520280589166210, 6.98396464687216219598665428081, 7.970922414955782464239708014317, 8.502339747541824544278461919128
