L(s) = 1 | + 2-s + 4-s + 5-s + 7-s + 8-s + 10-s + 6·13-s + 14-s + 16-s + 2·17-s + 20-s + 25-s + 6·26-s + 28-s + 6·29-s + 8·31-s + 32-s + 2·34-s + 35-s − 10·37-s + 40-s + 2·41-s − 4·43-s − 8·47-s + 49-s + 50-s + 6·52-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s + 0.353·8-s + 0.316·10-s + 1.66·13-s + 0.267·14-s + 1/4·16-s + 0.485·17-s + 0.223·20-s + 1/5·25-s + 1.17·26-s + 0.188·28-s + 1.11·29-s + 1.43·31-s + 0.176·32-s + 0.342·34-s + 0.169·35-s − 1.64·37-s + 0.158·40-s + 0.312·41-s − 0.609·43-s − 1.16·47-s + 1/7·49-s + 0.141·50-s + 0.832·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(6.574867646\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.574867646\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−13.97043358906687, −13.63667973080404, −13.14689114316789, −12.67527237540845, −11.95622925770598, −11.71960594585965, −11.09280755562849, −10.59970792590299, −10.12973207419958, −9.684059455402721, −8.771597812518727, −8.434278492704306, −8.061616007653127, −7.230192147009678, −6.572176443941464, −6.345489685802262, −5.676126902324592, −5.107386404610967, −4.697271204490248, −3.839649662895920, −3.496659158261299, −2.786589236652486, −2.069269226195511, −1.361932488100788, −0.8036202257214043,
0.8036202257214043, 1.361932488100788, 2.069269226195511, 2.786589236652486, 3.496659158261299, 3.839649662895920, 4.697271204490248, 5.107386404610967, 5.676126902324592, 6.345489685802262, 6.572176443941464, 7.230192147009678, 8.061616007653127, 8.434278492704306, 8.771597812518727, 9.684059455402721, 10.12973207419958, 10.59970792590299, 11.09280755562849, 11.71960594585965, 11.95622925770598, 12.67527237540845, 13.14689114316789, 13.63667973080404, 13.97043358906687