| L(s) = 1 | + 2-s + 2.16·3-s + 4-s − 5-s + 2.16·6-s − 2.30·7-s + 8-s + 1.69·9-s − 10-s + 2.61·11-s + 2.16·12-s − 2.15·13-s − 2.30·14-s − 2.16·15-s + 16-s + 1.42·17-s + 1.69·18-s + 1.22·19-s − 20-s − 4.99·21-s + 2.61·22-s + 0.562·23-s + 2.16·24-s + 25-s − 2.15·26-s − 2.83·27-s − 2.30·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.25·3-s + 0.5·4-s − 0.447·5-s + 0.884·6-s − 0.872·7-s + 0.353·8-s + 0.564·9-s − 0.316·10-s + 0.787·11-s + 0.625·12-s − 0.598·13-s − 0.616·14-s − 0.559·15-s + 0.250·16-s + 0.346·17-s + 0.399·18-s + 0.280·19-s − 0.223·20-s − 1.09·21-s + 0.556·22-s + 0.117·23-s + 0.442·24-s + 0.200·25-s − 0.423·26-s − 0.544·27-s − 0.436·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.411807462\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.411807462\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 601 | \( 1 + T \) |
| good | 3 | \( 1 - 2.16T + 3T^{2} \) |
| 7 | \( 1 + 2.30T + 7T^{2} \) |
| 11 | \( 1 - 2.61T + 11T^{2} \) |
| 13 | \( 1 + 2.15T + 13T^{2} \) |
| 17 | \( 1 - 1.42T + 17T^{2} \) |
| 19 | \( 1 - 1.22T + 19T^{2} \) |
| 23 | \( 1 - 0.562T + 23T^{2} \) |
| 29 | \( 1 + 0.528T + 29T^{2} \) |
| 31 | \( 1 - 10.4T + 31T^{2} \) |
| 37 | \( 1 - 6.47T + 37T^{2} \) |
| 41 | \( 1 + 2.23T + 41T^{2} \) |
| 43 | \( 1 - 4.25T + 43T^{2} \) |
| 47 | \( 1 - 12.1T + 47T^{2} \) |
| 53 | \( 1 + 0.489T + 53T^{2} \) |
| 59 | \( 1 - 14.2T + 59T^{2} \) |
| 61 | \( 1 + 1.31T + 61T^{2} \) |
| 67 | \( 1 + 8.93T + 67T^{2} \) |
| 71 | \( 1 - 11.4T + 71T^{2} \) |
| 73 | \( 1 - 13.4T + 73T^{2} \) |
| 79 | \( 1 - 12.9T + 79T^{2} \) |
| 83 | \( 1 + 5.32T + 83T^{2} \) |
| 89 | \( 1 + 13.6T + 89T^{2} \) |
| 97 | \( 1 - 7.58T + 97T^{2} \) |
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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
−8.009283142266642567715211878706, −7.40425270942779442961250897801, −6.68881360060834709951800526232, −6.04190982208256480378600894455, −5.06090960621433158410512622292, −4.12253633292903717893882337880, −3.65755134596809208202130891715, −2.85230347716673097148545197032, −2.35252811478532288175127465421, −0.942524513778396941851279164709,
0.942524513778396941851279164709, 2.35252811478532288175127465421, 2.85230347716673097148545197032, 3.65755134596809208202130891715, 4.12253633292903717893882337880, 5.06090960621433158410512622292, 6.04190982208256480378600894455, 6.68881360060834709951800526232, 7.40425270942779442961250897801, 8.009283142266642567715211878706
