| L(s) = 1 | − 4.20·2-s − 3·3-s + 9.71·4-s + 12.6·6-s + 11.2·7-s − 7.22·8-s + 9·9-s + 25.8·11-s − 29.1·12-s − 13·13-s − 47.3·14-s − 47.3·16-s + 20.3·17-s − 37.8·18-s + 154.·19-s − 33.7·21-s − 108.·22-s + 180.·23-s + 21.6·24-s + 54.7·26-s − 27·27-s + 109.·28-s − 20.4·29-s + 266.·31-s + 256.·32-s − 77.6·33-s − 85.5·34-s + ⋯ |
| L(s) = 1 | − 1.48·2-s − 0.577·3-s + 1.21·4-s + 0.859·6-s + 0.607·7-s − 0.319·8-s + 0.333·9-s + 0.709·11-s − 0.701·12-s − 0.277·13-s − 0.904·14-s − 0.739·16-s + 0.290·17-s − 0.496·18-s + 1.86·19-s − 0.350·21-s − 1.05·22-s + 1.63·23-s + 0.184·24-s + 0.412·26-s − 0.192·27-s + 0.738·28-s − 0.130·29-s + 1.54·31-s + 1.41·32-s − 0.409·33-s − 0.431·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.017142058\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.017142058\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + 13T \) |
| good | 2 | \( 1 + 4.20T + 8T^{2} \) |
| 7 | \( 1 - 11.2T + 343T^{2} \) |
| 11 | \( 1 - 25.8T + 1.33e3T^{2} \) |
| 17 | \( 1 - 20.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 154.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 180.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 20.4T + 2.43e4T^{2} \) |
| 31 | \( 1 - 266.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 115.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 391.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 151.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 467.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 79.9T + 1.48e5T^{2} \) |
| 59 | \( 1 + 873.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 187.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 609.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 248.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 852.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 331.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 435.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 259.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.22e3T + 9.12e5T^{2} \) |
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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
−9.494178574889453449532725566230, −9.012020058158586110211477915930, −7.926982445869481575030714698757, −7.35401239567934309439535138866, −6.54394677745459173042049709959, −5.35218635141876564078536102752, −4.47709681510470457546332003594, −2.93128026712554258580104114065, −1.42494093046967229782767446772, −0.802573242418448795591162097164,
0.802573242418448795591162097164, 1.42494093046967229782767446772, 2.93128026712554258580104114065, 4.47709681510470457546332003594, 5.35218635141876564078536102752, 6.54394677745459173042049709959, 7.35401239567934309439535138866, 7.926982445869481575030714698757, 9.012020058158586110211477915930, 9.494178574889453449532725566230
