| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s − 2.92·7-s + 8-s + 9-s + 0.190·11-s − 12-s − 0.310·13-s − 2.92·14-s + 16-s − 6.04·17-s + 18-s + 3.23·19-s + 2.92·21-s + 0.190·22-s − 4.47·23-s − 24-s − 0.310·26-s − 27-s − 2.92·28-s + 6.81·29-s − 1.69·31-s + 32-s − 0.190·33-s − 6.04·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.408·6-s − 1.10·7-s + 0.353·8-s + 0.333·9-s + 0.0573·11-s − 0.288·12-s − 0.0859·13-s − 0.782·14-s + 0.250·16-s − 1.46·17-s + 0.235·18-s + 0.742·19-s + 0.638·21-s + 0.0405·22-s − 0.932·23-s − 0.204·24-s − 0.0608·26-s − 0.192·27-s − 0.553·28-s + 1.26·29-s − 0.303·31-s + 0.176·32-s − 0.0331·33-s − 1.03·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.900857095\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.900857095\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 + 2.92T + 7T^{2} \) |
| 11 | \( 1 - 0.190T + 11T^{2} \) |
| 13 | \( 1 + 0.310T + 13T^{2} \) |
| 17 | \( 1 + 6.04T + 17T^{2} \) |
| 19 | \( 1 - 3.23T + 19T^{2} \) |
| 23 | \( 1 + 4.47T + 23T^{2} \) |
| 29 | \( 1 - 6.81T + 29T^{2} \) |
| 31 | \( 1 + 1.69T + 31T^{2} \) |
| 37 | \( 1 - 9.75T + 37T^{2} \) |
| 41 | \( 1 - 5.07T + 41T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 47 | \( 1 - 10.7T + 47T^{2} \) |
| 53 | \( 1 + 9.97T + 53T^{2} \) |
| 59 | \( 1 - 7.42T + 59T^{2} \) |
| 61 | \( 1 + 5.54T + 61T^{2} \) |
| 67 | \( 1 - 15.0T + 67T^{2} \) |
| 71 | \( 1 - 6.85T + 71T^{2} \) |
| 73 | \( 1 + 11.6T + 73T^{2} \) |
| 79 | \( 1 - 1.57T + 79T^{2} \) |
| 83 | \( 1 + 13.0T + 83T^{2} \) |
| 89 | \( 1 + 9.28T + 89T^{2} \) |
| 97 | \( 1 + 1.73T + 97T^{2} \) |
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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
−8.478650524781573919486283580735, −7.48250963862956513260223168254, −6.84566374486949161007670124423, −6.12948579067348350442650719616, −5.72588288672056724278056191451, −4.56013428145610897912324457738, −4.10980112453575596757600651156, −3.02430891941891840023118308040, −2.24210311114820638125546752923, −0.72214120844711079340142348256,
0.72214120844711079340142348256, 2.24210311114820638125546752923, 3.02430891941891840023118308040, 4.10980112453575596757600651156, 4.56013428145610897912324457738, 5.72588288672056724278056191451, 6.12948579067348350442650719616, 6.84566374486949161007670124423, 7.48250963862956513260223168254, 8.478650524781573919486283580735
