L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s + 7-s − 8-s + 9-s + 10-s − 2·11-s − 12-s + 13-s − 14-s + 15-s + 16-s + 3·17-s − 18-s − 2·19-s − 20-s − 21-s + 2·22-s + 3·23-s + 24-s + 25-s − 26-s − 27-s + 28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.603·11-s − 0.288·12-s + 0.277·13-s − 0.267·14-s + 0.258·15-s + 1/4·16-s + 0.727·17-s − 0.235·18-s − 0.458·19-s − 0.223·20-s − 0.218·21-s + 0.426·22-s + 0.625·23-s + 0.204·24-s + 1/5·25-s − 0.196·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 327990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 327990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 29 | \( 1 \) |
good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 13 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 + 15 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 13 T + p T^{2} \) |
| 97 | \( 1 - 3 T + p T^{2} \) |
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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
−12.80515938075992, −12.22191236592489, −11.81251984060294, −11.33376172356746, −11.12723022824066, −10.49359425794492, −10.17726381109676, −9.772119777261882, −9.164703354014615, −8.589212613879790, −8.216913949985512, −7.890002495233733, −7.281792538684807, −6.841498470675255, −6.481982967616197, −5.754981082306418, −5.357551204973252, −4.897832406834887, −4.335315211551798, −3.673205887020479, −3.150320839315829, −2.617127646008381, −1.758311205807791, −1.411108088979703, −0.6194812573570457, 0,
0.6194812573570457, 1.411108088979703, 1.758311205807791, 2.617127646008381, 3.150320839315829, 3.673205887020479, 4.335315211551798, 4.897832406834887, 5.357551204973252, 5.754981082306418, 6.481982967616197, 6.841498470675255, 7.281792538684807, 7.890002495233733, 8.216913949985512, 8.589212613879790, 9.164703354014615, 9.772119777261882, 10.17726381109676, 10.49359425794492, 11.12723022824066, 11.33376172356746, 11.81251984060294, 12.22191236592489, 12.80515938075992