| L(s) = 1 | − 2-s − 3-s − 2·4-s + 6-s − 7-s + 3·8-s + 9-s + 2·11-s + 2·12-s − 4·13-s + 14-s + 16-s + 3·17-s − 18-s − 6·19-s + 21-s − 2·22-s − 6·23-s − 3·24-s − 5·25-s + 4·26-s − 27-s + 2·28-s − 5·29-s − 6·31-s − 2·32-s − 2·33-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 4-s + 0.408·6-s − 0.377·7-s + 1.06·8-s + 1/3·9-s + 0.603·11-s + 0.577·12-s − 1.10·13-s + 0.267·14-s + 1/4·16-s + 0.727·17-s − 0.235·18-s − 1.37·19-s + 0.218·21-s − 0.426·22-s − 1.25·23-s − 0.612·24-s − 25-s + 0.784·26-s − 0.192·27-s + 0.377·28-s − 0.928·29-s − 1.07·31-s − 0.353·32-s − 0.348·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10017 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10017 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( 1 + T \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 2 T + p T^{2} ) \) |
| 53 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 5 T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 T - 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 17 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 3 T + 7 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 6 T + 24 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 6 T + 22 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 5 T + 16 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T - 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 5 T + 34 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 3 T + 13 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 3 T - 13 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 11 T + 67 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 18 T + 171 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 53 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 7 T + 107 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 5 T + 103 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 4 T + 116 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 6 T + 123 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $D_{4}$ | \( 1 + 6 T + 118 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−17.0617294532, −16.5041922801, −16.0948966824, −15.4294619348, −14.6792556283, −14.5081423238, −13.8607363749, −13.2910226366, −12.7341704170, −12.4043809095, −11.7538733675, −11.2507554675, −10.4323824738, −9.87410954624, −9.76893499415, −9.04245859330, −8.49473856256, −7.87025220617, −7.24042127650, −6.51665666696, −5.71934591949, −5.13679485501, −4.19893283968, −3.74326782327, −2.00830817157, 0,
2.00830817157, 3.74326782327, 4.19893283968, 5.13679485501, 5.71934591949, 6.51665666696, 7.24042127650, 7.87025220617, 8.49473856256, 9.04245859330, 9.76893499415, 9.87410954624, 10.4323824738, 11.2507554675, 11.7538733675, 12.4043809095, 12.7341704170, 13.2910226366, 13.8607363749, 14.5081423238, 14.6792556283, 15.4294619348, 16.0948966824, 16.5041922801, 17.0617294532
