| L(s) = 1 | + 0.414·2-s − 3.41·3-s − 1.82·4-s − 5-s − 1.41·6-s − 0.585·7-s − 1.58·8-s + 8.65·9-s − 0.414·10-s − 2.58·11-s + 6.24·12-s − 2.82·13-s − 0.242·14-s + 3.41·15-s + 3·16-s − 17-s + 3.58·18-s − 2.82·19-s + 1.82·20-s + 2·21-s − 1.07·22-s − 3.41·23-s + 5.41·24-s + 25-s − 1.17·26-s − 19.3·27-s + 1.07·28-s + ⋯ |
| L(s) = 1 | + 0.292·2-s − 1.97·3-s − 0.914·4-s − 0.447·5-s − 0.577·6-s − 0.221·7-s − 0.560·8-s + 2.88·9-s − 0.130·10-s − 0.779·11-s + 1.80·12-s − 0.784·13-s − 0.0648·14-s + 0.881·15-s + 0.750·16-s − 0.242·17-s + 0.845·18-s − 0.648·19-s + 0.408·20-s + 0.436·21-s − 0.228·22-s − 0.711·23-s + 1.10·24-s + 0.200·25-s − 0.229·26-s − 3.71·27-s + 0.202·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 2 | \( 1 - 0.414T + 2T^{2} \) |
| 3 | \( 1 + 3.41T + 3T^{2} \) |
| 7 | \( 1 + 0.585T + 7T^{2} \) |
| 11 | \( 1 + 2.58T + 11T^{2} \) |
| 13 | \( 1 + 2.82T + 13T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 + 3.41T + 23T^{2} \) |
| 29 | \( 1 + 4.82T + 29T^{2} \) |
| 31 | \( 1 - 4.24T + 31T^{2} \) |
| 37 | \( 1 - 6.48T + 37T^{2} \) |
| 41 | \( 1 + 6.48T + 41T^{2} \) |
| 43 | \( 1 - 7.65T + 43T^{2} \) |
| 47 | \( 1 + 4.82T + 47T^{2} \) |
| 53 | \( 1 - 0.343T + 53T^{2} \) |
| 59 | \( 1 + 9.17T + 59T^{2} \) |
| 61 | \( 1 - 7.65T + 61T^{2} \) |
| 67 | \( 1 + 3.17T + 67T^{2} \) |
| 71 | \( 1 + 4.24T + 71T^{2} \) |
| 73 | \( 1 + 4.82T + 73T^{2} \) |
| 79 | \( 1 - 5.41T + 79T^{2} \) |
| 83 | \( 1 - 9.31T + 83T^{2} \) |
| 89 | \( 1 + 2.34T + 89T^{2} \) |
| 97 | \( 1 - 3.65T + 97T^{2} \) |
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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
−13.26182409131366112701738937553, −12.57805276161533657208570884782, −11.72015210452327744400488944927, −10.57847569920422134851528053543, −9.653658474315122243342761418134, −7.75778713540626957529223558073, −6.31406013723778519305150017524, −5.19016061384923701683556479987, −4.27247643895843496960279170082, 0,
4.27247643895843496960279170082, 5.19016061384923701683556479987, 6.31406013723778519305150017524, 7.75778713540626957529223558073, 9.653658474315122243342761418134, 10.57847569920422134851528053543, 11.72015210452327744400488944927, 12.57805276161533657208570884782, 13.26182409131366112701738937553
