| L(s) = 1 | + 18.2·5-s + (18.5 − 0.256i)7-s + 56.9i·11-s − 10.8i·13-s − 65.2·17-s + 36.6i·19-s − 81.1i·23-s + 207.·25-s + 269. i·29-s + 135. i·31-s + (337. − 4.68i)35-s − 125.·37-s − 235.·41-s − 45.2·43-s − 482.·47-s + ⋯ |
| L(s) = 1 | + 1.63·5-s + (0.999 − 0.0138i)7-s + 1.56i·11-s − 0.230i·13-s − 0.931·17-s + 0.441i·19-s − 0.735i·23-s + 1.66·25-s + 1.72i·29-s + 0.787i·31-s + (1.63 − 0.0226i)35-s − 0.558·37-s − 0.895·41-s − 0.160·43-s − 1.49·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0138 - 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0138 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.053047484\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.053047484\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-18.5 + 0.256i)T \) |
| good | 5 | \( 1 - 18.2T + 125T^{2} \) |
| 11 | \( 1 - 56.9iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 10.8iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 65.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 36.6iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 81.1iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 269. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 135. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 125.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 235.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 45.2T + 7.95e4T^{2} \) |
| 47 | \( 1 + 482.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 70.1iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 353.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 591. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 523.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 895. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 982. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 1.02e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 304.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.02e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 782. iT - 9.12e5T^{2} \) |
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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
−8.840185813375835656328922909570, −8.303759996021388132976584425044, −6.97795192169381979358614919498, −6.77739093576186290673686033514, −5.49535907168593642970937573751, −5.06549296285958867550693221710, −4.25702089998939934263233119805, −2.76899098023293251850476351998, −1.87357853330781628584390242403, −1.42546626526603529036831034414,
0.52420219930930969124867274695, 1.70790371610354737661856541628, 2.31306958607233608161250045114, 3.45739088110513633086388747235, 4.67700233876042248416423055592, 5.39240745211233875468696151184, 6.09664300707957233975327311915, 6.66431195303285913933555850547, 7.891862059831666064646854802590, 8.535367254838267060281443660557
