| L(s) = 1 | + 4.14·2-s + 3.50·3-s + 9.16·4-s + 1.63·5-s + 14.5·6-s + 7·7-s + 4.81·8-s − 14.6·9-s + 6.75·10-s + 30.3·11-s + 32.1·12-s − 33.2·13-s + 28.9·14-s + 5.72·15-s − 53.3·16-s − 60.8·18-s − 96.5·19-s + 14.9·20-s + 24.5·21-s + 125.·22-s − 210.·23-s + 16.8·24-s − 122.·25-s − 137.·26-s − 146.·27-s + 64.1·28-s + 143.·29-s + ⋯ |
| L(s) = 1 | + 1.46·2-s + 0.675·3-s + 1.14·4-s + 0.145·5-s + 0.988·6-s + 0.377·7-s + 0.212·8-s − 0.544·9-s + 0.213·10-s + 0.832·11-s + 0.773·12-s − 0.710·13-s + 0.553·14-s + 0.0985·15-s − 0.833·16-s − 0.796·18-s − 1.16·19-s + 0.167·20-s + 0.255·21-s + 1.21·22-s − 1.90·23-s + 0.143·24-s − 0.978·25-s − 1.04·26-s − 1.04·27-s + 0.432·28-s + 0.916·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2023 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 - 7T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 4.14T + 8T^{2} \) |
| 3 | \( 1 - 3.50T + 27T^{2} \) |
| 5 | \( 1 - 1.63T + 125T^{2} \) |
| 11 | \( 1 - 30.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 33.2T + 2.19e3T^{2} \) |
| 19 | \( 1 + 96.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 210.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 143.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 47.2T + 2.97e4T^{2} \) |
| 37 | \( 1 + 207.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 338.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 454.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 335.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 222.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 213.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 128.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 610.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 723.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.14e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 473.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 577.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 444.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 98.9T + 9.12e5T^{2} \) |
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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
−8.384155313448338683107547516028, −7.55273815534565876421483302943, −6.48434485949982324960385479217, −5.94257010414841309055557884014, −5.06775651920352746786693591160, −4.12720198402743783873172787443, −3.66261639024753598030119488526, −2.48866147210261724787077237257, −1.94516046385955544166226730636, 0,
1.94516046385955544166226730636, 2.48866147210261724787077237257, 3.66261639024753598030119488526, 4.12720198402743783873172787443, 5.06775651920352746786693591160, 5.94257010414841309055557884014, 6.48434485949982324960385479217, 7.55273815534565876421483302943, 8.384155313448338683107547516028
