| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s + 11-s − 12-s − 2·13-s + 16-s + 5·17-s − 18-s + 8·19-s − 22-s + 8·23-s + 24-s − 5·25-s + 2·26-s − 27-s − 2·29-s + 31-s − 32-s − 33-s − 5·34-s + 36-s − 5·37-s − 8·38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.301·11-s − 0.288·12-s − 0.554·13-s + 1/4·16-s + 1.21·17-s − 0.235·18-s + 1.83·19-s − 0.213·22-s + 1.66·23-s + 0.204·24-s − 25-s + 0.392·26-s − 0.192·27-s − 0.371·29-s + 0.179·31-s − 0.176·32-s − 0.174·33-s − 0.857·34-s + 1/6·36-s − 0.821·37-s − 1.29·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100254 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100254 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.063508681\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.063508681\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 13 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−13.64549826477173, −13.32156831718766, −12.52475483337463, −12.13126126469308, −11.85057096921429, −11.15728550594026, −10.95322046367564, −10.15452393331202, −9.837770159991591, −9.269448700579452, −9.087344284650949, −8.075460469214976, −7.749167092961270, −7.220539270474656, −6.897788821227214, −6.120296353667006, −5.525833997150449, −5.257659260001451, −4.585361849182189, −3.661202254789226, −3.300657542236091, −2.540199297995157, −1.795722848376833, −0.9250737947792183, −0.7054465974150164,
0.7054465974150164, 0.9250737947792183, 1.795722848376833, 2.540199297995157, 3.300657542236091, 3.661202254789226, 4.585361849182189, 5.257659260001451, 5.525833997150449, 6.120296353667006, 6.897788821227214, 7.220539270474656, 7.749167092961270, 8.075460469214976, 9.087344284650949, 9.269448700579452, 9.837770159991591, 10.15452393331202, 10.95322046367564, 11.15728550594026, 11.85057096921429, 12.13126126469308, 12.52475483337463, 13.32156831718766, 13.64549826477173
