| L(s) = 1 | + 2.23·2-s + 3-s + 3.00·4-s + 2·5-s + 2.23·6-s − 4.47·7-s + 2.23·8-s + 9-s + 4.47·10-s + 3.00·12-s − 10.0·14-s + 2·15-s − 0.999·16-s + 4.47·17-s + 2.23·18-s − 4.47·19-s + 6.00·20-s − 4.47·21-s − 4·23-s + 2.23·24-s − 25-s + 27-s − 13.4·28-s + 4.47·29-s + 4.47·30-s − 6.70·32-s + 10.0·34-s + ⋯ |
| L(s) = 1 | + 1.58·2-s + 0.577·3-s + 1.50·4-s + 0.894·5-s + 0.912·6-s − 1.69·7-s + 0.790·8-s + 0.333·9-s + 1.41·10-s + 0.866·12-s − 2.67·14-s + 0.516·15-s − 0.249·16-s + 1.08·17-s + 0.527·18-s − 1.02·19-s + 1.34·20-s − 0.975·21-s − 0.834·23-s + 0.456·24-s − 0.200·25-s + 0.192·27-s − 2.53·28-s + 0.830·29-s + 0.816·30-s − 1.18·32-s + 1.71·34-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
| \(L(1)\) |
\(\approx\) |
\(3.456194644\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.456194644\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.23T + 2T^{2} \) |
| 5 | \( 1 - 2T + 5T^{2} \) |
| 7 | \( 1 + 4.47T + 7T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 4.47T + 17T^{2} \) |
| 19 | \( 1 + 4.47T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 4.47T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + 4.47T + 41T^{2} \) |
| 43 | \( 1 - 4.47T + 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 8.94T + 61T^{2} \) |
| 67 | \( 1 + 12T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 8.94T + 73T^{2} \) |
| 79 | \( 1 - 13.4T + 79T^{2} \) |
| 83 | \( 1 + 8.94T + 83T^{2} \) |
| 89 | \( 1 + 14T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−11.96813214338545798912435604191, −10.38104632610696553786721885962, −9.800274452758984125650845855443, −8.768845110335529202437101245554, −7.21034934891176342175331080954, −6.21714992398416050933747042136, −5.73387068866665816679127942391, −4.23324492548299928706192072988, −3.27526656858308043035087270177, −2.34514472825920689329196705733,
2.34514472825920689329196705733, 3.27526656858308043035087270177, 4.23324492548299928706192072988, 5.73387068866665816679127942391, 6.21714992398416050933747042136, 7.21034934891176342175331080954, 8.768845110335529202437101245554, 9.800274452758984125650845855443, 10.38104632610696553786721885962, 11.96813214338545798912435604191
