| L(s) = 1 | − 2·2-s − 3-s + 2·4-s + 5-s + 2·6-s − 7-s + 9-s − 2·10-s − 3·11-s − 2·12-s − 6·13-s + 2·14-s − 15-s − 4·16-s − 2·18-s − 19-s + 2·20-s + 21-s + 6·22-s + 25-s + 12·26-s − 27-s − 2·28-s + 5·29-s + 2·30-s + 8·31-s + 8·32-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s + 4-s + 0.447·5-s + 0.816·6-s − 0.377·7-s + 1/3·9-s − 0.632·10-s − 0.904·11-s − 0.577·12-s − 1.66·13-s + 0.534·14-s − 0.258·15-s − 16-s − 0.471·18-s − 0.229·19-s + 0.447·20-s + 0.218·21-s + 1.27·22-s + 1/5·25-s + 2.35·26-s − 0.192·27-s − 0.377·28-s + 0.928·29-s + 0.365·30-s + 1.43·31-s + 1.41·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30345 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30345 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−15.82811389524017, −15.02720785053160, −14.37956293949175, −13.76625582416581, −13.14711888908856, −12.70801935519858, −11.98521499075363, −11.63827345037075, −10.84489705090803, −10.23778783520398, −10.03158476905236, −9.724236041040383, −8.838937694199849, −8.455273480926633, −7.709924738173199, −7.263305664197827, −6.729194407768085, −6.133075114238305, −5.295675232669858, −4.826206619671803, −4.207943298934284, −2.872086778937287, −2.502711506421783, −1.664564381407011, −0.7185349995431105, 0,
0.7185349995431105, 1.664564381407011, 2.502711506421783, 2.872086778937287, 4.207943298934284, 4.826206619671803, 5.295675232669858, 6.133075114238305, 6.729194407768085, 7.263305664197827, 7.709924738173199, 8.455273480926633, 8.838937694199849, 9.724236041040383, 10.03158476905236, 10.23778783520398, 10.84489705090803, 11.63827345037075, 11.98521499075363, 12.70801935519858, 13.14711888908856, 13.76625582416581, 14.37956293949175, 15.02720785053160, 15.82811389524017
