| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 4.07·7-s + 8-s + 9-s + 10-s + 12-s + 4.13·13-s + 4.07·14-s + 15-s + 16-s − 5.97·17-s + 18-s + 5.35·19-s + 20-s + 4.07·21-s + 3.33·23-s + 24-s + 25-s + 4.13·26-s + 27-s + 4.07·28-s − 2.45·29-s + 30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s + 1.53·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s + 0.288·12-s + 1.14·13-s + 1.08·14-s + 0.258·15-s + 0.250·16-s − 1.44·17-s + 0.235·18-s + 1.22·19-s + 0.223·20-s + 0.888·21-s + 0.695·23-s + 0.204·24-s + 0.200·25-s + 0.811·26-s + 0.192·27-s + 0.769·28-s − 0.455·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.171826562\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.171826562\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 - 4.07T + 7T^{2} \) |
| 13 | \( 1 - 4.13T + 13T^{2} \) |
| 17 | \( 1 + 5.97T + 17T^{2} \) |
| 19 | \( 1 - 5.35T + 19T^{2} \) |
| 23 | \( 1 - 3.33T + 23T^{2} \) |
| 29 | \( 1 + 2.45T + 29T^{2} \) |
| 31 | \( 1 + 4.92T + 31T^{2} \) |
| 37 | \( 1 - 2.14T + 37T^{2} \) |
| 41 | \( 1 + 7.10T + 41T^{2} \) |
| 43 | \( 1 + 11.6T + 43T^{2} \) |
| 47 | \( 1 + 4.01T + 47T^{2} \) |
| 53 | \( 1 + 1.63T + 53T^{2} \) |
| 59 | \( 1 + 6.56T + 59T^{2} \) |
| 61 | \( 1 - 4.14T + 61T^{2} \) |
| 67 | \( 1 - 13.0T + 67T^{2} \) |
| 71 | \( 1 + 13.1T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 - 4.97T + 79T^{2} \) |
| 83 | \( 1 + 10.4T + 83T^{2} \) |
| 89 | \( 1 - 13.5T + 89T^{2} \) |
| 97 | \( 1 - 8.43T + 97T^{2} \) |
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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
−8.556395553711409913128575956119, −7.80312603148641764409522432587, −7.04635219345799579453242310746, −6.28708461230981249303864779872, −5.26885953468035508265110201691, −4.84599003598359735233965934743, −3.90850334231113997084688882607, −3.08735183534069079670389209856, −1.96136402672513403221459647420, −1.40087884158235982011218936620,
1.40087884158235982011218936620, 1.96136402672513403221459647420, 3.08735183534069079670389209856, 3.90850334231113997084688882607, 4.84599003598359735233965934743, 5.26885953468035508265110201691, 6.28708461230981249303864779872, 7.04635219345799579453242310746, 7.80312603148641764409522432587, 8.556395553711409913128575956119
