| L(s) = 1 | + (2 + 2i)2-s + 8i·4-s + (−2 − 11i)5-s + (−16 + 16i)8-s − 27i·9-s + (18 − 26i)10-s + (−37 + 37i)13-s − 64·16-s + (99 + 99i)17-s + (54 − 54i)18-s + (88 − 16i)20-s + (−117 + 44i)25-s − 148·26-s − 284i·29-s + (−128 − 128i)32-s + ⋯ |
| L(s) = 1 | + (0.707 + 0.707i)2-s + i·4-s + (−0.178 − 0.983i)5-s + (−0.707 + 0.707i)8-s − i·9-s + (0.569 − 0.822i)10-s + (−0.789 + 0.789i)13-s − 16-s + (1.41 + 1.41i)17-s + (0.707 − 0.707i)18-s + (0.983 − 0.178i)20-s + (−0.936 + 0.351i)25-s − 1.11·26-s − 1.81i·29-s + (−0.707 − 0.707i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.742 - 0.669i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.742 - 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.31007 + 0.503184i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.31007 + 0.503184i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-2 - 2i)T \) |
| 5 | \( 1 + (2 + 11i)T \) |
| good | 3 | \( 1 + 27iT^{2} \) |
| 7 | \( 1 - 343iT^{2} \) |
| 11 | \( 1 - 1.33e3T^{2} \) |
| 13 | \( 1 + (37 - 37i)T - 2.19e3iT^{2} \) |
| 17 | \( 1 + (-99 - 99i)T + 4.91e3iT^{2} \) |
| 19 | \( 1 + 6.85e3T^{2} \) |
| 23 | \( 1 + 1.21e4iT^{2} \) |
| 29 | \( 1 + 284iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 2.97e4T^{2} \) |
| 37 | \( 1 + (91 + 91i)T + 5.06e4iT^{2} \) |
| 41 | \( 1 - 472T + 6.89e4T^{2} \) |
| 43 | \( 1 + 7.95e4iT^{2} \) |
| 47 | \( 1 - 1.03e5iT^{2} \) |
| 53 | \( 1 + (27 - 27i)T - 1.48e5iT^{2} \) |
| 59 | \( 1 + 2.05e5T^{2} \) |
| 61 | \( 1 + 468T + 2.26e5T^{2} \) |
| 67 | \( 1 - 3.00e5iT^{2} \) |
| 71 | \( 1 - 3.57e5T^{2} \) |
| 73 | \( 1 + (-253 + 253i)T - 3.89e5iT^{2} \) |
| 79 | \( 1 + 4.93e5T^{2} \) |
| 83 | \( 1 + 5.71e5iT^{2} \) |
| 89 | \( 1 - 176iT - 7.04e5T^{2} \) |
| 97 | \( 1 + (611 + 611i)T + 9.12e5iT^{2} \) |
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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
−17.45546087292726170577777845224, −16.69752237222435590939438327202, −15.38727699628562890512432007082, −14.27626125064912458928709606258, −12.69500147627342730624120166724, −11.94950548885687097205042938930, −9.345497442084030485072288587476, −7.83807535551204722256362042230, −5.95265495077236844226719802653, −4.12937926458756835599315736874,
2.92881682390944771765176714559, 5.26244609920122227016028362377, 7.36530271200470390972058886539, 9.946969841461214799388836065548, 10.99742868321278542442602360451, 12.34280805707255941107442787719, 13.88166295641359620815192458007, 14.74714739455204021924409479449, 16.17593689634869811986342933635, 18.14684558954369962380112005112
