L(s) = 1 | − 2-s − 3-s + 4-s + 6-s + 4·7-s − 8-s + 9-s + 4·11-s − 12-s + 6·13-s − 4·14-s + 16-s − 2·17-s − 18-s − 19-s − 4·21-s − 4·22-s − 8·23-s + 24-s − 6·26-s − 27-s + 4·28-s − 2·29-s − 4·31-s − 32-s − 4·33-s + 2·34-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 1.51·7-s − 0.353·8-s + 1/3·9-s + 1.20·11-s − 0.288·12-s + 1.66·13-s − 1.06·14-s + 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.229·19-s − 0.872·21-s − 0.852·22-s − 1.66·23-s + 0.204·24-s − 1.17·26-s − 0.192·27-s + 0.755·28-s − 0.371·29-s − 0.718·31-s − 0.176·32-s − 0.696·33-s + 0.342·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.028268787\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.028268787\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 19 | \( 1 + T \) |
| 43 | \( 1 - T \) |
good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 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
−13.50204454505234, −13.13528347167775, −12.32050683610339, −11.87556912792902, −11.40409304652294, −11.28916888549697, −10.69729886864205, −10.27334072772545, −9.631090226838861, −9.039570277541214, −8.602897993033867, −8.212538179957580, −7.763327552924218, −7.098027283323189, −6.492748943995798, −6.169930899482061, −5.591532418401192, −5.020702916246250, −4.206175996958086, −3.967989476656348, −3.311341093659974, −2.091937776958619, −1.777191846250223, −1.274872261725936, −0.5355394826947360,
0.5355394826947360, 1.274872261725936, 1.777191846250223, 2.091937776958619, 3.311341093659974, 3.967989476656348, 4.206175996958086, 5.020702916246250, 5.591532418401192, 6.169930899482061, 6.492748943995798, 7.098027283323189, 7.763327552924218, 8.212538179957580, 8.602897993033867, 9.039570277541214, 9.631090226838861, 10.27334072772545, 10.69729886864205, 11.28916888549697, 11.40409304652294, 11.87556912792902, 12.32050683610339, 13.13528347167775, 13.50204454505234