| L(s) = 1 | − 7.63·3-s + 7.65·5-s − 5.64·7-s + 31.3·9-s + 64.9·11-s − 54.0·13-s − 58.4·15-s − 20.0·17-s − 52.3·19-s + 43.1·21-s + 132.·23-s − 66.4·25-s − 33.1·27-s + 29·29-s − 135.·31-s − 496.·33-s − 43.1·35-s + 137.·37-s + 412.·39-s − 229.·41-s + 46.2·43-s + 239.·45-s − 327.·47-s − 311.·49-s + 153.·51-s − 32.9·53-s + 496.·55-s + ⋯ |
| L(s) = 1 | − 1.46·3-s + 0.684·5-s − 0.304·7-s + 1.16·9-s + 1.78·11-s − 1.15·13-s − 1.00·15-s − 0.286·17-s − 0.632·19-s + 0.448·21-s + 1.19·23-s − 0.531·25-s − 0.236·27-s + 0.185·29-s − 0.782·31-s − 2.61·33-s − 0.208·35-s + 0.612·37-s + 1.69·39-s − 0.874·41-s + 0.163·43-s + 0.794·45-s − 1.01·47-s − 0.907·49-s + 0.421·51-s − 0.0853·53-s + 1.21·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 29 | \( 1 - 29T \) |
| good | 3 | \( 1 + 7.63T + 27T^{2} \) |
| 5 | \( 1 - 7.65T + 125T^{2} \) |
| 7 | \( 1 + 5.64T + 343T^{2} \) |
| 11 | \( 1 - 64.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 54.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 20.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 52.3T + 6.85e3T^{2} \) |
| 23 | \( 1 - 132.T + 1.21e4T^{2} \) |
| 31 | \( 1 + 135.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 137.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 229.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 46.2T + 7.95e4T^{2} \) |
| 47 | \( 1 + 327.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 32.9T + 1.48e5T^{2} \) |
| 59 | \( 1 - 378.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 18.5T + 2.26e5T^{2} \) |
| 67 | \( 1 - 998.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 570.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 224.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 430.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.39e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 131.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 837.T + 9.12e5T^{2} \) |
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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
−8.701592458479374874535701111707, −7.34705047266050327516110287633, −6.53537158556285716839560194590, −6.30008779335521136952947223019, −5.25859688410240496722789173889, −4.65413394308345009474492164782, −3.58977842123380133426444576809, −2.14753571347409760857434373270, −1.11108465961738298836708632840, 0,
1.11108465961738298836708632840, 2.14753571347409760857434373270, 3.58977842123380133426444576809, 4.65413394308345009474492164782, 5.25859688410240496722789173889, 6.30008779335521136952947223019, 6.53537158556285716839560194590, 7.34705047266050327516110287633, 8.701592458479374874535701111707
