| L(s) = 1 | + 2·2-s + 4·4-s − 6.40·5-s + 27.0·7-s + 8·8-s − 12.8·10-s − 41.6·13-s + 54.1·14-s + 16·16-s − 72.1·17-s + 14.2·19-s − 25.6·20-s + 24.8·23-s − 83.9·25-s − 83.3·26-s + 108.·28-s + 254.·29-s − 46.6·31-s + 32·32-s − 144.·34-s − 173.·35-s + 263.·37-s + 28.4·38-s − 51.2·40-s + 447.·41-s − 35.0·43-s + 49.7·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.572·5-s + 1.46·7-s + 0.353·8-s − 0.405·10-s − 0.889·13-s + 1.03·14-s + 0.250·16-s − 1.02·17-s + 0.171·19-s − 0.286·20-s + 0.225·23-s − 0.671·25-s − 0.628·26-s + 0.730·28-s + 1.62·29-s − 0.270·31-s + 0.176·32-s − 0.727·34-s − 0.836·35-s + 1.16·37-s + 0.121·38-s − 0.202·40-s + 1.70·41-s − 0.124·43-s + 0.159·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2178 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2178 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.734754886\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.734754886\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 6.40T + 125T^{2} \) |
| 7 | \( 1 - 27.0T + 343T^{2} \) |
| 13 | \( 1 + 41.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 72.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 14.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 24.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 254.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 46.6T + 2.97e4T^{2} \) |
| 37 | \( 1 - 263.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 447.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 35.0T + 7.95e4T^{2} \) |
| 47 | \( 1 - 476.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 97.9T + 1.48e5T^{2} \) |
| 59 | \( 1 - 189.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 448.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 383.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 631.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 920.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 690.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 221.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 921.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 208.T + 9.12e5T^{2} \) |
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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
−8.521153501201905461309028392034, −7.77085912397717214476699806555, −7.30175645041993670249850699943, −6.28236973292834105891606328699, −5.34436354361256445407083111394, −4.51282098262395920449677942209, −4.20546304734267885653916002096, −2.80419760894072328493591452643, −2.03013284390767361703392721490, −0.789554879536812991014805929664,
0.789554879536812991014805929664, 2.03013284390767361703392721490, 2.80419760894072328493591452643, 4.20546304734267885653916002096, 4.51282098262395920449677942209, 5.34436354361256445407083111394, 6.28236973292834105891606328699, 7.30175645041993670249850699943, 7.77085912397717214476699806555, 8.521153501201905461309028392034
