L(s) = 1 | + (0.763 + 2.72i)2-s + (5.19 + 0.151i)3-s + (−6.83 + 4.16i)4-s + (4.66 + 4.66i)5-s + (3.55 + 14.2i)6-s + 0.405·7-s + (−16.5 − 15.4i)8-s + (26.9 + 1.56i)9-s + (−9.14 + 16.2i)10-s + (−5.82 + 5.82i)11-s + (−36.1 + 20.5i)12-s + (−35.2 − 35.2i)13-s + (0.309 + 1.10i)14-s + (23.5 + 24.9i)15-s + (29.3 − 56.8i)16-s + 49.3i·17-s + ⋯ |
L(s) = 1 | + (0.270 + 0.962i)2-s + (0.999 + 0.0290i)3-s + (−0.854 + 0.520i)4-s + (0.417 + 0.417i)5-s + (0.241 + 0.970i)6-s + 0.0219·7-s + (−0.731 − 0.681i)8-s + (0.998 + 0.0581i)9-s + (−0.289 + 0.514i)10-s + (−0.159 + 0.159i)11-s + (−0.868 + 0.495i)12-s + (−0.751 − 0.751i)13-s + (0.00591 + 0.0210i)14-s + (0.405 + 0.429i)15-s + (0.459 − 0.888i)16-s + 0.703i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.113 - 0.993i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.113 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.43504 + 1.28100i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.43504 + 1.28100i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.763 - 2.72i)T \) |
| 3 | \( 1 + (-5.19 - 0.151i)T \) |
good | 5 | \( 1 + (-4.66 - 4.66i)T + 125iT^{2} \) |
| 7 | \( 1 - 0.405T + 343T^{2} \) |
| 11 | \( 1 + (5.82 - 5.82i)T - 1.33e3iT^{2} \) |
| 13 | \( 1 + (35.2 + 35.2i)T + 2.19e3iT^{2} \) |
| 17 | \( 1 - 49.3iT - 4.91e3T^{2} \) |
| 19 | \( 1 + (-108. + 108. i)T - 6.85e3iT^{2} \) |
| 23 | \( 1 + 130. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + (172. - 172. i)T - 2.43e4iT^{2} \) |
| 31 | \( 1 - 36.1iT - 2.97e4T^{2} \) |
| 37 | \( 1 + (257. - 257. i)T - 5.06e4iT^{2} \) |
| 41 | \( 1 + 5.87T + 6.89e4T^{2} \) |
| 43 | \( 1 + (-170. - 170. i)T + 7.95e4iT^{2} \) |
| 47 | \( 1 + 181.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (148. + 148. i)T + 1.48e5iT^{2} \) |
| 59 | \( 1 + (567. - 567. i)T - 2.05e5iT^{2} \) |
| 61 | \( 1 + (-481. - 481. i)T + 2.26e5iT^{2} \) |
| 67 | \( 1 + (-296. + 296. i)T - 3.00e5iT^{2} \) |
| 71 | \( 1 + 533. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 178. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 528. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + (-713. - 713. i)T + 5.71e5iT^{2} \) |
| 89 | \( 1 - 204.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 275.T + 9.12e5T^{2} \) |
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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
−15.16902894525375141160640717429, −14.48023476841495496730781549707, −13.47507299234697229039207680141, −12.50979865239024153449626360788, −10.28012444601289635050442313633, −9.130197991610662616759271146920, −7.87379324788344464940872202504, −6.75995357207590190693217319255, −4.93131154801195101403233031225, −3.03857469378377731092670161126,
1.83924860026765039050600761976, 3.59878182753573292425028081027, 5.29899900323881701655813865662, 7.66976681548948934636675557704, 9.296706571919606244939240079119, 9.774635163476749470622413052037, 11.53242225668048921272015530535, 12.70296566421590242772368270959, 13.76587364992118821626843736824, 14.38510423192457414873443321408