| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 8-s + 9-s + 10-s − 12-s + 15-s + 16-s + 17-s − 18-s − 2·19-s − 20-s + 24-s + 25-s − 27-s + 4·29-s − 30-s − 8·31-s − 32-s − 34-s + 36-s − 2·37-s + 2·38-s + 40-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s + 0.258·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s − 0.458·19-s − 0.223·20-s + 0.204·24-s + 1/5·25-s − 0.192·27-s + 0.742·29-s − 0.182·30-s − 1.43·31-s − 0.176·32-s − 0.171·34-s + 1/6·36-s − 0.328·37-s + 0.324·38-s + 0.158·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.89276683211038, −15.14850592236136, −14.73721457364061, −14.18238634033654, −13.29309444215880, −12.90240815052926, −12.15391212480989, −11.89233727496954, −11.14400122629828, −10.84223453526841, −10.17235923144179, −9.716826575849604, −9.013834495299792, −8.423391260160898, −7.956977002722391, −7.212428357711490, −6.793530602837545, −6.191190203635170, −5.403139475518122, −4.937831741941744, −3.993287757287928, −3.489486529324676, −2.528406687084007, −1.753326475733632, −0.8614831159411040, 0,
0.8614831159411040, 1.753326475733632, 2.528406687084007, 3.489486529324676, 3.993287757287928, 4.937831741941744, 5.403139475518122, 6.191190203635170, 6.793530602837545, 7.212428357711490, 7.956977002722391, 8.423391260160898, 9.013834495299792, 9.716826575849604, 10.17235923144179, 10.84223453526841, 11.14400122629828, 11.89233727496954, 12.15391212480989, 12.90240815052926, 13.29309444215880, 14.18238634033654, 14.73721457364061, 15.14850592236136, 15.89276683211038
