| L(s) = 1 | + 3.68·2-s + 5.94·3-s + 5.54·4-s − 5·5-s + 21.8·6-s + 18.5·7-s − 9.02·8-s + 8.34·9-s − 18.4·10-s − 9.43·11-s + 32.9·12-s − 13·13-s + 68.3·14-s − 29.7·15-s − 77.6·16-s + 20.8·17-s + 30.7·18-s − 156.·19-s − 27.7·20-s + 110.·21-s − 34.7·22-s − 145.·23-s − 53.6·24-s + 25·25-s − 47.8·26-s − 110.·27-s + 102.·28-s + ⋯ |
| L(s) = 1 | + 1.30·2-s + 1.14·3-s + 0.693·4-s − 0.447·5-s + 1.48·6-s + 1.00·7-s − 0.398·8-s + 0.309·9-s − 0.581·10-s − 0.258·11-s + 0.793·12-s − 0.277·13-s + 1.30·14-s − 0.511·15-s − 1.21·16-s + 0.298·17-s + 0.402·18-s − 1.89·19-s − 0.310·20-s + 1.14·21-s − 0.336·22-s − 1.32·23-s − 0.456·24-s + 0.200·25-s − 0.360·26-s − 0.790·27-s + 0.694·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + 5T \) |
| 13 | \( 1 + 13T \) |
| 31 | \( 1 - 31T \) |
| good | 2 | \( 1 - 3.68T + 8T^{2} \) |
| 3 | \( 1 - 5.94T + 27T^{2} \) |
| 7 | \( 1 - 18.5T + 343T^{2} \) |
| 11 | \( 1 + 9.43T + 1.33e3T^{2} \) |
| 17 | \( 1 - 20.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 156.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 145.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 111.T + 2.43e4T^{2} \) |
| 37 | \( 1 + 33.3T + 5.06e4T^{2} \) |
| 41 | \( 1 + 24.5T + 6.89e4T^{2} \) |
| 43 | \( 1 - 116.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 590.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 49.7T + 1.48e5T^{2} \) |
| 59 | \( 1 - 492.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 254.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 1.07e3T + 3.00e5T^{2} \) |
| 71 | \( 1 - 134.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 124.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 87.5T + 4.93e5T^{2} \) |
| 83 | \( 1 - 572.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 589.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 406.T + 9.12e5T^{2} \) |
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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
−8.289352431261659450336686584240, −7.83942042805708067241253600142, −6.70910498179266691251673221511, −5.85871221801453404443506003661, −4.85548327522002944490119129844, −4.26829243592275812069615421961, −3.53839913503253264812802166110, −2.58994700812610047419058556875, −1.88852906511639908309209360893, 0,
1.88852906511639908309209360893, 2.58994700812610047419058556875, 3.53839913503253264812802166110, 4.26829243592275812069615421961, 4.85548327522002944490119129844, 5.85871221801453404443506003661, 6.70910498179266691251673221511, 7.83942042805708067241253600142, 8.289352431261659450336686584240
