| L(s) = 1 | − 2-s + 3-s + 4-s − 0.203·5-s − 6-s + 0.561·7-s − 8-s + 9-s + 0.203·10-s − 2.27·11-s + 12-s − 13-s − 0.561·14-s − 0.203·15-s + 16-s − 3.49·17-s − 18-s − 2.88·19-s − 0.203·20-s + 0.561·21-s + 2.27·22-s + 1.20·23-s − 24-s − 4.95·25-s + 26-s + 27-s + 0.561·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.0911·5-s − 0.408·6-s + 0.212·7-s − 0.353·8-s + 0.333·9-s + 0.0644·10-s − 0.686·11-s + 0.288·12-s − 0.277·13-s − 0.150·14-s − 0.0526·15-s + 0.250·16-s − 0.847·17-s − 0.235·18-s − 0.660·19-s − 0.0455·20-s + 0.122·21-s + 0.485·22-s + 0.251·23-s − 0.204·24-s − 0.991·25-s + 0.196·26-s + 0.192·27-s + 0.106·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.402662472\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.402662472\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 + T \) |
| good | 5 | \( 1 + 0.203T + 5T^{2} \) |
| 7 | \( 1 - 0.561T + 7T^{2} \) |
| 11 | \( 1 + 2.27T + 11T^{2} \) |
| 17 | \( 1 + 3.49T + 17T^{2} \) |
| 19 | \( 1 + 2.88T + 19T^{2} \) |
| 23 | \( 1 - 1.20T + 23T^{2} \) |
| 29 | \( 1 - 4.71T + 29T^{2} \) |
| 31 | \( 1 - 3.74T + 31T^{2} \) |
| 37 | \( 1 + 2.70T + 37T^{2} \) |
| 41 | \( 1 - 0.285T + 41T^{2} \) |
| 43 | \( 1 - 4.78T + 43T^{2} \) |
| 47 | \( 1 + 0.764T + 47T^{2} \) |
| 53 | \( 1 - 12.3T + 53T^{2} \) |
| 59 | \( 1 - 5.47T + 59T^{2} \) |
| 61 | \( 1 + 12.0T + 61T^{2} \) |
| 67 | \( 1 + 0.176T + 67T^{2} \) |
| 71 | \( 1 + 4.55T + 71T^{2} \) |
| 73 | \( 1 - 8.20T + 73T^{2} \) |
| 79 | \( 1 + 7.70T + 79T^{2} \) |
| 83 | \( 1 - 15.6T + 83T^{2} \) |
| 89 | \( 1 - 18.2T + 89T^{2} \) |
| 97 | \( 1 - 7.19T + 97T^{2} \) |
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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
−7.966666081513158748747408503984, −7.33048002038290289085617193899, −6.63008789337175390770594870811, −5.93014947928889237544023350652, −4.93370137821883325701493383451, −4.29028987027298994535102053787, −3.31812046416630796128213724893, −2.45916147552663413519575117233, −1.90625390543814326101909030140, −0.61219951525887575070897328044,
0.61219951525887575070897328044, 1.90625390543814326101909030140, 2.45916147552663413519575117233, 3.31812046416630796128213724893, 4.29028987027298994535102053787, 4.93370137821883325701493383451, 5.93014947928889237544023350652, 6.63008789337175390770594870811, 7.33048002038290289085617193899, 7.966666081513158748747408503984
