| L(s) = 1 | − 4.92·2-s − 4.99·3-s + 16.2·4-s − 2.12·5-s + 24.5·6-s − 29.0·7-s − 40.6·8-s − 2.08·9-s + 10.4·10-s + 23.0·11-s − 81.1·12-s + 63.7·13-s + 143.·14-s + 10.5·15-s + 70.3·16-s − 17.2·17-s + 10.2·18-s − 142.·19-s − 34.5·20-s + 145.·21-s − 113.·22-s − 38.3·23-s + 203.·24-s − 120.·25-s − 314.·26-s + 145.·27-s − 472.·28-s + ⋯ |
| L(s) = 1 | − 1.74·2-s − 0.960·3-s + 2.03·4-s − 0.189·5-s + 1.67·6-s − 1.56·7-s − 1.79·8-s − 0.0772·9-s + 0.330·10-s + 0.631·11-s − 1.95·12-s + 1.36·13-s + 2.73·14-s + 0.182·15-s + 1.09·16-s − 0.245·17-s + 0.134·18-s − 1.71·19-s − 0.386·20-s + 1.50·21-s − 1.10·22-s − 0.347·23-s + 1.72·24-s − 0.963·25-s − 2.37·26-s + 1.03·27-s − 3.18·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 43 | \( 1 \) |
| good | 2 | \( 1 + 4.92T + 8T^{2} \) |
| 3 | \( 1 + 4.99T + 27T^{2} \) |
| 5 | \( 1 + 2.12T + 125T^{2} \) |
| 7 | \( 1 + 29.0T + 343T^{2} \) |
| 11 | \( 1 - 23.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 63.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 17.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 142.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 38.3T + 1.21e4T^{2} \) |
| 29 | \( 1 - 217.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 240.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 27.2T + 5.06e4T^{2} \) |
| 41 | \( 1 - 175.T + 6.89e4T^{2} \) |
| 47 | \( 1 - 25.7T + 1.03e5T^{2} \) |
| 53 | \( 1 + 195.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 781.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 695.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 29.0T + 3.00e5T^{2} \) |
| 71 | \( 1 + 941.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 236.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 171.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 318.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 837.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 665.T + 9.12e5T^{2} \) |
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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
−8.800916969691819444625692383416, −7.87487473698392898677452932058, −6.76249361190255293769624720795, −6.33377796574558106002692053128, −5.92704343464095142718414953549, −4.22274793692677999242049736546, −3.20624633320979359294261352974, −1.95888704476563250908199748789, −0.72244864130515011988803020726, 0,
0.72244864130515011988803020726, 1.95888704476563250908199748789, 3.20624633320979359294261352974, 4.22274793692677999242049736546, 5.92704343464095142718414953549, 6.33377796574558106002692053128, 6.76249361190255293769624720795, 7.87487473698392898677452932058, 8.800916969691819444625692383416
