| L(s) = 1 | − 3.33·2-s + 3·3-s + 3.12·4-s + 0.253·5-s − 10.0·6-s − 0.691·7-s + 16.2·8-s + 9·9-s − 0.844·10-s + 11·11-s + 9.36·12-s + 22.1·13-s + 2.30·14-s + 0.759·15-s − 79.2·16-s + 110.·17-s − 30.0·18-s + 32.5·19-s + 0.790·20-s − 2.07·21-s − 36.6·22-s − 160.·23-s + 48.8·24-s − 124.·25-s − 73.8·26-s + 27·27-s − 2.15·28-s + ⋯ |
| L(s) = 1 | − 1.17·2-s + 0.577·3-s + 0.390·4-s + 0.0226·5-s − 0.680·6-s − 0.0373·7-s + 0.718·8-s + 0.333·9-s − 0.0266·10-s + 0.301·11-s + 0.225·12-s + 0.472·13-s + 0.0440·14-s + 0.0130·15-s − 1.23·16-s + 1.57·17-s − 0.393·18-s + 0.393·19-s + 0.00883·20-s − 0.0215·21-s − 0.355·22-s − 1.45·23-s + 0.415·24-s − 0.999·25-s − 0.556·26-s + 0.192·27-s − 0.0145·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 3T \) |
| 11 | \( 1 - 11T \) |
| 61 | \( 1 - 61T \) |
| good | 2 | \( 1 + 3.33T + 8T^{2} \) |
| 5 | \( 1 - 0.253T + 125T^{2} \) |
| 7 | \( 1 + 0.691T + 343T^{2} \) |
| 13 | \( 1 - 22.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 110.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 32.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 160.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 40.8T + 2.43e4T^{2} \) |
| 31 | \( 1 + 207.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 109.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 179.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 108.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 284.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 414.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 170.T + 2.05e5T^{2} \) |
| 67 | \( 1 + 497.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 257.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.17e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 736.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 637.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 349.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 758.T + 9.12e5T^{2} \) |
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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
−8.422502157131511430507210427729, −7.83094021657400944632632218945, −7.28075154990962889363280548660, −6.19055667438381302804136775587, −5.26253777551506098170960117568, −4.06131419161782741346493682112, −3.35796970648318201581192945712, −1.97163460138340609844197095364, −1.24710429175565633093324975291, 0,
1.24710429175565633093324975291, 1.97163460138340609844197095364, 3.35796970648318201581192945712, 4.06131419161782741346493682112, 5.26253777551506098170960117568, 6.19055667438381302804136775587, 7.28075154990962889363280548660, 7.83094021657400944632632218945, 8.422502157131511430507210427729
