| L(s) = 1 | + 1.22·2-s − 6.48·4-s − 17.3·5-s + 19.1·7-s − 17.8·8-s − 21.2·10-s + 32.6·11-s + 23.5·14-s + 30.0·16-s − 86.5·17-s − 76.0·19-s + 112.·20-s + 40.1·22-s + 75.3·23-s + 174.·25-s − 124.·28-s + 286.·29-s − 200.·31-s + 179.·32-s − 106.·34-s − 332.·35-s + 173.·37-s − 93.4·38-s + 308.·40-s + 496.·41-s − 187.·43-s − 211.·44-s + ⋯ |
| L(s) = 1 | + 0.434·2-s − 0.811·4-s − 1.54·5-s + 1.03·7-s − 0.786·8-s − 0.672·10-s + 0.895·11-s + 0.450·14-s + 0.469·16-s − 1.23·17-s − 0.918·19-s + 1.25·20-s + 0.388·22-s + 0.682·23-s + 1.39·25-s − 0.840·28-s + 1.83·29-s − 1.16·31-s + 0.990·32-s − 0.536·34-s − 1.60·35-s + 0.769·37-s − 0.399·38-s + 1.21·40-s + 1.88·41-s − 0.664·43-s − 0.726·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 1.22T + 8T^{2} \) |
| 5 | \( 1 + 17.3T + 125T^{2} \) |
| 7 | \( 1 - 19.1T + 343T^{2} \) |
| 11 | \( 1 - 32.6T + 1.33e3T^{2} \) |
| 17 | \( 1 + 86.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 76.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 75.3T + 1.21e4T^{2} \) |
| 29 | \( 1 - 286.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 200.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 173.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 496.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 187.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 254.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 63.6T + 1.48e5T^{2} \) |
| 59 | \( 1 + 67.9T + 2.05e5T^{2} \) |
| 61 | \( 1 - 400.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 510.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 200.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 168.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.11e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 724.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 821.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 19.5T + 9.12e5T^{2} \) |
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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
−8.674511580332020716331987232150, −8.045483836046964751251651168406, −7.16077925662222323142324825626, −6.24157922046015317520584609230, −5.00825296672786621215682938623, −4.25869720659889489846584328906, −4.04339847728795365137030353984, −2.73122761614940507139521672273, −1.11306478795608159847115203144, 0,
1.11306478795608159847115203144, 2.73122761614940507139521672273, 4.04339847728795365137030353984, 4.25869720659889489846584328906, 5.00825296672786621215682938623, 6.24157922046015317520584609230, 7.16077925662222323142324825626, 8.045483836046964751251651168406, 8.674511580332020716331987232150
