| L(s) = 1 | − 1.67·2-s − 3.03·3-s − 5.20·4-s + 19.6·5-s + 5.06·6-s − 28.4·7-s + 22.0·8-s − 17.8·9-s − 32.7·10-s − 28.1·11-s + 15.7·12-s + 20.8·13-s + 47.5·14-s − 59.4·15-s + 4.72·16-s + 24.2·17-s + 29.7·18-s − 118.·19-s − 102.·20-s + 86.2·21-s + 47.0·22-s + 61.3·23-s − 66.9·24-s + 259.·25-s − 34.8·26-s + 135.·27-s + 148.·28-s + ⋯ |
| L(s) = 1 | − 0.591·2-s − 0.583·3-s − 0.650·4-s + 1.75·5-s + 0.344·6-s − 1.53·7-s + 0.975·8-s − 0.659·9-s − 1.03·10-s − 0.771·11-s + 0.379·12-s + 0.445·13-s + 0.908·14-s − 1.02·15-s + 0.0738·16-s + 0.345·17-s + 0.389·18-s − 1.43·19-s − 1.14·20-s + 0.896·21-s + 0.456·22-s + 0.556·23-s − 0.569·24-s + 2.07·25-s − 0.263·26-s + 0.968·27-s + 0.999·28-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 | \( 1 + 1.67T + 8T^{2} \) |
| 3 | \( 1 + 3.03T + 27T^{2} \) |
| 5 | \( 1 - 19.6T + 125T^{2} \) |
| 7 | \( 1 + 28.4T + 343T^{2} \) |
| 11 | \( 1 + 28.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 20.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 24.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 118.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 61.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + 70.4T + 2.43e4T^{2} \) |
| 31 | \( 1 - 263.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 360.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 67.4T + 6.89e4T^{2} \) |
| 47 | \( 1 + 457.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 148.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 182.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 215.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 631.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 575.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.11e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 622.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 475.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.39e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.06e3T + 9.12e5T^{2} \) |
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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
−8.788153108904880899257249034840, −7.899495075208955194909163189649, −6.43677662953373306517062374518, −6.28095734144367623574781815196, −5.43733005992649965368676730850, −4.61888243361928004557869169140, −3.18577747814817816636310990846, −2.33157098224038683356958342355, −0.976469312884327350551761672063, 0,
0.976469312884327350551761672063, 2.33157098224038683356958342355, 3.18577747814817816636310990846, 4.61888243361928004557869169140, 5.43733005992649965368676730850, 6.28095734144367623574781815196, 6.43677662953373306517062374518, 7.899495075208955194909163189649, 8.788153108904880899257249034840
