| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 8-s + 9-s + 4·11-s + 12-s − 2·13-s + 16-s + 17-s + 18-s − 2·19-s + 4·22-s + 2·23-s + 24-s − 5·25-s − 2·26-s + 27-s + 32-s + 4·33-s + 34-s + 36-s − 10·37-s − 2·38-s − 2·39-s + 6·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.353·8-s + 1/3·9-s + 1.20·11-s + 0.288·12-s − 0.554·13-s + 1/4·16-s + 0.242·17-s + 0.235·18-s − 0.458·19-s + 0.852·22-s + 0.417·23-s + 0.204·24-s − 25-s − 0.392·26-s + 0.192·27-s + 0.176·32-s + 0.696·33-s + 0.171·34-s + 1/6·36-s − 1.64·37-s − 0.324·38-s − 0.320·39-s + 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 85782 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85782 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 29 | \( 1 \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.15648689159313, −13.81934131451848, −13.15251356681820, −12.79517094044909, −12.21480569852963, −11.81355543715387, −11.33883750063669, −10.77962407473169, −10.10971399809487, −9.668445520615932, −9.233813875702180, −8.553275383939926, −8.134582962533927, −7.500957166411896, −6.893562401626579, −6.598736270678926, −5.945639450573595, −5.250239357626414, −4.805642428599418, −3.990804357094016, −3.783338010261492, −3.088008671517432, −2.411132230070175, −1.771872292990342, −1.196633009123010, 0,
1.196633009123010, 1.771872292990342, 2.411132230070175, 3.088008671517432, 3.783338010261492, 3.990804357094016, 4.805642428599418, 5.250239357626414, 5.945639450573595, 6.598736270678926, 6.893562401626579, 7.500957166411896, 8.134582962533927, 8.553275383939926, 9.233813875702180, 9.668445520615932, 10.10971399809487, 10.77962407473169, 11.33883750063669, 11.81355543715387, 12.21480569852963, 12.79517094044909, 13.15251356681820, 13.81934131451848, 14.15648689159313
