| L(s) = 1 | − 2.56·2-s + 3·3-s − 1.43·4-s − 7.68·6-s − 29.1·7-s + 24.1·8-s + 9·9-s − 11·11-s − 4.31·12-s + 39.1·13-s + 74.5·14-s − 50.4·16-s − 46.3·17-s − 23.0·18-s + 43.5·19-s − 87.3·21-s + 28.1·22-s − 91.3·23-s + 72.5·24-s − 100.·26-s + 27·27-s + 41.8·28-s − 185.·29-s − 45.8·31-s − 64.2·32-s − 33·33-s + 118.·34-s + ⋯ |
| L(s) = 1 | − 0.905·2-s + 0.577·3-s − 0.179·4-s − 0.522·6-s − 1.57·7-s + 1.06·8-s + 0.333·9-s − 0.301·11-s − 0.103·12-s + 0.835·13-s + 1.42·14-s − 0.787·16-s − 0.661·17-s − 0.301·18-s + 0.526·19-s − 0.907·21-s + 0.273·22-s − 0.827·23-s + 0.616·24-s − 0.756·26-s + 0.192·27-s + 0.282·28-s − 1.19·29-s − 0.265·31-s − 0.354·32-s − 0.174·33-s + 0.598·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.7984545134\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7984545134\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 + 2.56T + 8T^{2} \) |
| 7 | \( 1 + 29.1T + 343T^{2} \) |
| 13 | \( 1 - 39.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 46.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 43.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 91.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + 185.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 45.8T + 2.97e4T^{2} \) |
| 37 | \( 1 + 177.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 232.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 64.1T + 7.95e4T^{2} \) |
| 47 | \( 1 + 16.9T + 1.03e5T^{2} \) |
| 53 | \( 1 - 48.7T + 1.48e5T^{2} \) |
| 59 | \( 1 - 200.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 521.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 338.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 318.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.13e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 824.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 731.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 902.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.35e3T + 9.12e5T^{2} \) |
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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
−9.629419378391355068618997386217, −9.108696821228017947744083115866, −8.350037804315865952195649070751, −7.44704390762062069178973095411, −6.62440703219610203687963052713, −5.55346182672484139482005048643, −4.11367329747820484710185538312, −3.35082261123523780242825207840, −2.01455076993076590657450081882, −0.54312167352892343227909177845,
0.54312167352892343227909177845, 2.01455076993076590657450081882, 3.35082261123523780242825207840, 4.11367329747820484710185538312, 5.55346182672484139482005048643, 6.62440703219610203687963052713, 7.44704390762062069178973095411, 8.350037804315865952195649070751, 9.108696821228017947744083115866, 9.629419378391355068618997386217
