| L(s) = 1 | + (0.123 − 3.99i)2-s + (2.78 − 1.60i)3-s + (−15.9 − 0.987i)4-s + (16.3 − 28.3i)5-s + (−6.08 − 11.3i)6-s + (−41.5 − 25.9i)7-s + (−5.92 + 63.7i)8-s + (−35.3 + 61.1i)9-s + (−111. − 68.9i)10-s + (185. − 106. i)11-s + (−46.0 + 22.9i)12-s + 136.·13-s + (−108. + 162. i)14-s − 105. i·15-s + (254. + 31.5i)16-s + (138. + 240. i)17-s + ⋯ |
| L(s) = 1 | + (0.0308 − 0.999i)2-s + (0.309 − 0.178i)3-s + (−0.998 − 0.0617i)4-s + (0.654 − 1.13i)5-s + (−0.168 − 0.314i)6-s + (−0.848 − 0.529i)7-s + (−0.0925 + 0.995i)8-s + (−0.436 + 0.755i)9-s + (−1.11 − 0.689i)10-s + (1.53 − 0.884i)11-s + (−0.319 + 0.159i)12-s + 0.808·13-s + (−0.555 + 0.831i)14-s − 0.467i·15-s + (0.992 + 0.123i)16-s + (0.480 + 0.831i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.528 + 0.848i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.528 + 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.686371 - 1.23591i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.686371 - 1.23591i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.123 + 3.99i)T \) |
| 7 | \( 1 + (41.5 + 25.9i)T \) |
| good | 3 | \( 1 + (-2.78 + 1.60i)T + (40.5 - 70.1i)T^{2} \) |
| 5 | \( 1 + (-16.3 + 28.3i)T + (-312.5 - 541. i)T^{2} \) |
| 11 | \( 1 + (-185. + 106. i)T + (7.32e3 - 1.26e4i)T^{2} \) |
| 13 | \( 1 - 136.T + 2.85e4T^{2} \) |
| 17 | \( 1 + (-138. - 240. i)T + (-4.17e4 + 7.23e4i)T^{2} \) |
| 19 | \( 1 + (-103. - 59.9i)T + (6.51e4 + 1.12e5i)T^{2} \) |
| 23 | \( 1 + (132. + 76.4i)T + (1.39e5 + 2.42e5i)T^{2} \) |
| 29 | \( 1 + 383.T + 7.07e5T^{2} \) |
| 31 | \( 1 + (555. - 320. i)T + (4.61e5 - 7.99e5i)T^{2} \) |
| 37 | \( 1 + (-488. + 845. i)T + (-9.37e5 - 1.62e6i)T^{2} \) |
| 41 | \( 1 + 2.37e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + 474. iT - 3.41e6T^{2} \) |
| 47 | \( 1 + (-2.98e3 - 1.72e3i)T + (2.43e6 + 4.22e6i)T^{2} \) |
| 53 | \( 1 + (184. + 319. i)T + (-3.94e6 + 6.83e6i)T^{2} \) |
| 59 | \( 1 + (3.80e3 - 2.19e3i)T + (6.05e6 - 1.04e7i)T^{2} \) |
| 61 | \( 1 + (-1.00e3 + 1.74e3i)T + (-6.92e6 - 1.19e7i)T^{2} \) |
| 67 | \( 1 + (-265. + 153. i)T + (1.00e7 - 1.74e7i)T^{2} \) |
| 71 | \( 1 + 1.76e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + (-3.59e3 - 6.23e3i)T + (-1.41e7 + 2.45e7i)T^{2} \) |
| 79 | \( 1 + (-1.31e3 - 761. i)T + (1.94e7 + 3.37e7i)T^{2} \) |
| 83 | \( 1 - 9.69e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + (-846. + 1.46e3i)T + (-3.13e7 - 5.43e7i)T^{2} \) |
| 97 | \( 1 + 9.74e3T + 8.85e7T^{2} \) |
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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
−16.58305302356089018998662478442, −14.12205478345613795623559271410, −13.46619015896533688091091595770, −12.41488187820443883698055480801, −10.89000925497512449848086461140, −9.401602893443503749458662085017, −8.502127488134078550650028817293, −5.78222743432575203673083994750, −3.72682756622765492314544256564, −1.29964527528233466179560196032,
3.48007930921582713998912238139, 6.05426021644284289597948874339, 6.91606991090475100215349972350, 9.081966238731777139515521184312, 9.811498573794897242255491889913, 11.99850377151191935579915340137, 13.67814586370688143134711364895, 14.61004816515856767564798704641, 15.38926952556718935421073800712, 16.83670076570352538185254074057
