L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 7-s + 8-s + 9-s + 10-s + 5·11-s + 12-s − 13-s + 14-s + 15-s + 16-s − 3·17-s + 18-s − 8·19-s + 20-s + 21-s + 5·22-s − 4·23-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 + 1.50·11-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 − 1.83·19-s + 0.223·20-s + 0.218·21-s + 1.06·22-s − 0.834·23-s + 0.204·24-s + 1/5·25-s − 0.196·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37830 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37830 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(6.215091397\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.215091397\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 97 | \( 1 + T \) |
good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
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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
−14.66794330537699, −14.43767552699941, −13.92179623245040, −13.24982533098280, −12.95446755374710, −12.29645983709919, −11.83264957597524, −11.19486838059708, −10.76483098221908, −10.07557245744635, −9.520589623162203, −8.927867017961444, −8.505038824407182, −7.854257443388859, −7.163161760448861, −6.459034602056771, −6.299866850776167, −5.517647796659679, −4.656862106644139, −4.118439649779603, −3.914676257423867, −2.846952768575045, −2.162675950623610, −1.809540034683127, −0.7794062114562985,
0.7794062114562985, 1.809540034683127, 2.162675950623610, 2.846952768575045, 3.914676257423867, 4.118439649779603, 4.656862106644139, 5.517647796659679, 6.299866850776167, 6.459034602056771, 7.163161760448861, 7.854257443388859, 8.505038824407182, 8.927867017961444, 9.520589623162203, 10.07557245744635, 10.76483098221908, 11.19486838059708, 11.83264957597524, 12.29645983709919, 12.95446755374710, 13.24982533098280, 13.92179623245040, 14.43767552699941, 14.66794330537699