| L(s) = 1 | + 2.52·2-s + 3-s + 4.37·4-s − 2.37·5-s + 2.52·6-s + 0.792·7-s + 5.98·8-s + 9-s − 5.98·10-s + 4.37·12-s − 4.10·13-s + 2·14-s − 2.37·15-s + 6.37·16-s − 5.98·17-s + 2.52·18-s + 4.25·19-s − 10.3·20-s + 0.792·21-s + 2·23-s + 5.98·24-s + 0.627·25-s − 10.3·26-s + 27-s + 3.46·28-s − 2.52·29-s − 5.98·30-s + ⋯ |
| L(s) = 1 | + 1.78·2-s + 0.577·3-s + 2.18·4-s − 1.06·5-s + 1.03·6-s + 0.299·7-s + 2.11·8-s + 0.333·9-s − 1.89·10-s + 1.26·12-s − 1.13·13-s + 0.534·14-s − 0.612·15-s + 1.59·16-s − 1.45·17-s + 0.594·18-s + 0.976·19-s − 2.31·20-s + 0.172·21-s + 0.417·23-s + 1.22·24-s + 0.125·25-s − 2.03·26-s + 0.192·27-s + 0.654·28-s − 0.468·29-s − 1.09·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.614090370\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.614090370\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 2 | \( 1 - 2.52T + 2T^{2} \) |
| 5 | \( 1 + 2.37T + 5T^{2} \) |
| 7 | \( 1 - 0.792T + 7T^{2} \) |
| 13 | \( 1 + 4.10T + 13T^{2} \) |
| 17 | \( 1 + 5.98T + 17T^{2} \) |
| 19 | \( 1 - 4.25T + 19T^{2} \) |
| 23 | \( 1 - 2T + 23T^{2} \) |
| 29 | \( 1 + 2.52T + 29T^{2} \) |
| 31 | \( 1 - 7.37T + 31T^{2} \) |
| 37 | \( 1 - 5T + 37T^{2} \) |
| 41 | \( 1 + 5.69T + 41T^{2} \) |
| 43 | \( 1 + 6.63T + 43T^{2} \) |
| 47 | \( 1 + 1.25T + 47T^{2} \) |
| 53 | \( 1 - 13.1T + 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 + 2.67T + 61T^{2} \) |
| 67 | \( 1 - 16.1T + 67T^{2} \) |
| 71 | \( 1 - 0.744T + 71T^{2} \) |
| 73 | \( 1 - 7.42T + 73T^{2} \) |
| 79 | \( 1 - 5.84T + 79T^{2} \) |
| 83 | \( 1 - 8.51T + 83T^{2} \) |
| 89 | \( 1 + 6.37T + 89T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
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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
−11.70185077670714593076138147846, −11.07630728329445708473125184395, −9.718667796049862145367289080048, −8.301249964895047934689619514100, −7.37871115546445530999817545694, −6.62237116823775318658867605575, −5.09813228723920962325422278655, −4.42144199718635571856024854043, −3.42062086792945072410195055528, −2.33704734588710130923530548827,
2.33704734588710130923530548827, 3.42062086792945072410195055528, 4.42144199718635571856024854043, 5.09813228723920962325422278655, 6.62237116823775318658867605575, 7.37871115546445530999817545694, 8.301249964895047934689619514100, 9.718667796049862145367289080048, 11.07630728329445708473125184395, 11.70185077670714593076138147846
