| L(s) = 1 | + 1.59·2-s − 3-s + 0.539·4-s − 2.86·5-s − 1.59·6-s + 3.05·7-s − 2.32·8-s + 9-s − 4.56·10-s − 0.539·12-s − 13-s + 4.86·14-s + 2.86·15-s − 4.78·16-s + 0.406·17-s + 1.59·18-s + 0.593·19-s − 1.54·20-s − 3.05·21-s − 2.18·23-s + 2.32·24-s + 3.21·25-s − 1.59·26-s − 27-s + 1.64·28-s + 4.18·29-s + 4.56·30-s + ⋯ |
| L(s) = 1 | + 1.12·2-s − 0.577·3-s + 0.269·4-s − 1.28·5-s − 0.650·6-s + 1.15·7-s − 0.822·8-s + 0.333·9-s − 1.44·10-s − 0.155·12-s − 0.277·13-s + 1.30·14-s + 0.740·15-s − 1.19·16-s + 0.0985·17-s + 0.375·18-s + 0.136·19-s − 0.345·20-s − 0.666·21-s − 0.456·23-s + 0.475·24-s + 0.643·25-s − 0.312·26-s − 0.192·27-s + 0.311·28-s + 0.777·29-s + 0.834·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4719 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4719 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.798194177\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.798194177\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 11 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 - 1.59T + 2T^{2} \) |
| 5 | \( 1 + 2.86T + 5T^{2} \) |
| 7 | \( 1 - 3.05T + 7T^{2} \) |
| 17 | \( 1 - 0.406T + 17T^{2} \) |
| 19 | \( 1 - 0.593T + 19T^{2} \) |
| 23 | \( 1 + 2.18T + 23T^{2} \) |
| 29 | \( 1 - 4.18T + 29T^{2} \) |
| 31 | \( 1 + 3.73T + 31T^{2} \) |
| 37 | \( 1 + 1.14T + 37T^{2} \) |
| 41 | \( 1 - 4.94T + 41T^{2} \) |
| 43 | \( 1 + 4.73T + 43T^{2} \) |
| 47 | \( 1 - 3.78T + 47T^{2} \) |
| 53 | \( 1 - 6.59T + 53T^{2} \) |
| 59 | \( 1 + T + 59T^{2} \) |
| 61 | \( 1 + 15.0T + 61T^{2} \) |
| 67 | \( 1 - 13.4T + 67T^{2} \) |
| 71 | \( 1 - 5.48T + 71T^{2} \) |
| 73 | \( 1 - 14.6T + 73T^{2} \) |
| 79 | \( 1 + 15.0T + 79T^{2} \) |
| 83 | \( 1 - 12.4T + 83T^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 - 12.8T + 97T^{2} \) |
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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
−8.106385340744225871822418827203, −7.54686058801340117514396836349, −6.74614030350800276885856957796, −5.87336039534918237888246592727, −5.11175698402603379692749709580, −4.61525583744487586009468470499, −4.00091559404164556411655130479, −3.28167200334824216884444209876, −2.08090654256114820608143837899, −0.63677240245297213336501309325,
0.63677240245297213336501309325, 2.08090654256114820608143837899, 3.28167200334824216884444209876, 4.00091559404164556411655130479, 4.61525583744487586009468470499, 5.11175698402603379692749709580, 5.87336039534918237888246592727, 6.74614030350800276885856957796, 7.54686058801340117514396836349, 8.106385340744225871822418827203
