| L(s) = 1 | − 3.82·2-s + 6.65·4-s − 12.4·5-s + 5.14·8-s + 47.5·10-s + 39.4·11-s + 7.95·13-s − 72.9·16-s + 41.4·17-s − 112.·19-s − 82.6·20-s − 151.·22-s − 1.26·23-s + 29.1·25-s − 30.4·26-s − 37.1·29-s − 257.·31-s + 238.·32-s − 158.·34-s − 128.·37-s + 429.·38-s − 63.8·40-s − 130.·41-s + 493.·43-s + 262.·44-s + 4.85·46-s + 423.·47-s + ⋯ |
| L(s) = 1 | − 1.35·2-s + 0.832·4-s − 1.11·5-s + 0.227·8-s + 1.50·10-s + 1.08·11-s + 0.169·13-s − 1.13·16-s + 0.591·17-s − 1.35·19-s − 0.923·20-s − 1.46·22-s − 0.0114·23-s + 0.232·25-s − 0.229·26-s − 0.237·29-s − 1.49·31-s + 1.31·32-s − 0.800·34-s − 0.570·37-s + 1.83·38-s − 0.252·40-s − 0.498·41-s + 1.75·43-s + 0.900·44-s + 0.0155·46-s + 1.31·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 3.82T + 8T^{2} \) |
| 5 | \( 1 + 12.4T + 125T^{2} \) |
| 11 | \( 1 - 39.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 7.95T + 2.19e3T^{2} \) |
| 17 | \( 1 - 41.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 112.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 1.26T + 1.21e4T^{2} \) |
| 29 | \( 1 + 37.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + 257.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 128.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 130.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 493.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 423.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 577.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 294.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 67.5T + 2.26e5T^{2} \) |
| 67 | \( 1 - 987.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 715.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 340.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 81.6T + 4.93e5T^{2} \) |
| 83 | \( 1 + 846.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.31e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 715.T + 9.12e5T^{2} \) |
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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
−8.825598914480012675049488847398, −8.225301418840038481317918372500, −7.38208223884772472850539389718, −6.86058063298896491914217281851, −5.67431819133354653054082283609, −4.25101284896703264319687919152, −3.75639303810444641708480717361, −2.15682772641896408220025382330, −1.00912637869304958416781753381, 0,
1.00912637869304958416781753381, 2.15682772641896408220025382330, 3.75639303810444641708480717361, 4.25101284896703264319687919152, 5.67431819133354653054082283609, 6.86058063298896491914217281851, 7.38208223884772472850539389718, 8.225301418840038481317918372500, 8.825598914480012675049488847398
