L(s) = 1 | − 2-s + 0.994·3-s + 4-s − 3.18·5-s − 0.994·6-s − 0.721·7-s − 8-s − 2.01·9-s + 3.18·10-s − 1.88·11-s + 0.994·12-s + 1.86·13-s + 0.721·14-s − 3.17·15-s + 16-s + 2.19·17-s + 2.01·18-s − 19-s − 3.18·20-s − 0.717·21-s + 1.88·22-s + 3.75·23-s − 0.994·24-s + 5.16·25-s − 1.86·26-s − 4.98·27-s − 0.721·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.574·3-s + 0.5·4-s − 1.42·5-s − 0.406·6-s − 0.272·7-s − 0.353·8-s − 0.670·9-s + 1.00·10-s − 0.568·11-s + 0.287·12-s + 0.516·13-s + 0.192·14-s − 0.818·15-s + 0.250·16-s + 0.532·17-s + 0.473·18-s − 0.229·19-s − 0.712·20-s − 0.156·21-s + 0.402·22-s + 0.782·23-s − 0.203·24-s + 1.03·25-s − 0.365·26-s − 0.959·27-s − 0.136·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 + T \) |
good | 3 | \( 1 - 0.994T + 3T^{2} \) |
| 5 | \( 1 + 3.18T + 5T^{2} \) |
| 7 | \( 1 + 0.721T + 7T^{2} \) |
| 11 | \( 1 + 1.88T + 11T^{2} \) |
| 13 | \( 1 - 1.86T + 13T^{2} \) |
| 17 | \( 1 - 2.19T + 17T^{2} \) |
| 23 | \( 1 - 3.75T + 23T^{2} \) |
| 29 | \( 1 - 5.55T + 29T^{2} \) |
| 31 | \( 1 - 2.69T + 31T^{2} \) |
| 37 | \( 1 + 5.33T + 37T^{2} \) |
| 41 | \( 1 - 10.8T + 41T^{2} \) |
| 43 | \( 1 + 2.68T + 43T^{2} \) |
| 47 | \( 1 + 10.2T + 47T^{2} \) |
| 53 | \( 1 + 0.302T + 53T^{2} \) |
| 59 | \( 1 - 12.1T + 59T^{2} \) |
| 61 | \( 1 - 0.637T + 61T^{2} \) |
| 67 | \( 1 + 3.50T + 67T^{2} \) |
| 71 | \( 1 - 1.48T + 71T^{2} \) |
| 73 | \( 1 - 12.4T + 73T^{2} \) |
| 79 | \( 1 + 5.56T + 79T^{2} \) |
| 83 | \( 1 + 5.62T + 83T^{2} \) |
| 89 | \( 1 - 5.68T + 89T^{2} \) |
| 97 | \( 1 - 18.6T + 97T^{2} \) |
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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
−7.81493947446082763603023445566, −7.00532936527149352423487273985, −6.34840368937934946021842647325, −5.41398364432637087500176941174, −4.56651529253380421994428103262, −3.56594964096731115599252613352, −3.16505171426808491023315401011, −2.35429947166293418669597717488, −1.00133975522981844673510664130, 0,
1.00133975522981844673510664130, 2.35429947166293418669597717488, 3.16505171426808491023315401011, 3.56594964096731115599252613352, 4.56651529253380421994428103262, 5.41398364432637087500176941174, 6.34840368937934946021842647325, 7.00532936527149352423487273985, 7.81493947446082763603023445566