L(s) = 1 | + 2-s + (−0.707 − 0.707i)3-s + 4-s + (−2.23 − 0.132i)5-s + (−0.707 − 0.707i)6-s + (1.47 + 1.47i)7-s + 8-s + 1.00i·9-s + (−2.23 − 0.132i)10-s + 1.15i·11-s + (−0.707 − 0.707i)12-s − 2.88·13-s + (1.47 + 1.47i)14-s + (1.48 + 1.67i)15-s + 16-s + 5.62i·17-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (−0.408 − 0.408i)3-s + 0.5·4-s + (−0.998 − 0.0594i)5-s + (−0.288 − 0.288i)6-s + (0.559 + 0.559i)7-s + 0.353·8-s + 0.333i·9-s + (−0.705 − 0.0420i)10-s + 0.347i·11-s + (−0.204 − 0.204i)12-s − 0.801·13-s + (0.395 + 0.395i)14-s + (0.383 + 0.431i)15-s + 0.250·16-s + 1.36i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.265 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.265 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.476197360\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.476197360\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + (0.707 + 0.707i)T \) |
| 5 | \( 1 + (2.23 + 0.132i)T \) |
| 37 | \( 1 + (4.72 + 3.82i)T \) |
good | 7 | \( 1 + (-1.47 - 1.47i)T + 7iT^{2} \) |
| 11 | \( 1 - 1.15iT - 11T^{2} \) |
| 13 | \( 1 + 2.88T + 13T^{2} \) |
| 17 | \( 1 - 5.62iT - 17T^{2} \) |
| 19 | \( 1 + (1.32 - 1.32i)T - 19iT^{2} \) |
| 23 | \( 1 + 2.70T + 23T^{2} \) |
| 29 | \( 1 + (-6.59 - 6.59i)T + 29iT^{2} \) |
| 31 | \( 1 + (0.920 - 0.920i)T - 31iT^{2} \) |
| 41 | \( 1 - 6.84iT - 41T^{2} \) |
| 43 | \( 1 - 8.92T + 43T^{2} \) |
| 47 | \( 1 + (-5.84 - 5.84i)T + 47iT^{2} \) |
| 53 | \( 1 + (4.27 - 4.27i)T - 53iT^{2} \) |
| 59 | \( 1 + (7.37 - 7.37i)T - 59iT^{2} \) |
| 61 | \( 1 + (-0.721 + 0.721i)T - 61iT^{2} \) |
| 67 | \( 1 + (4.32 - 4.32i)T - 67iT^{2} \) |
| 71 | \( 1 + 4.66T + 71T^{2} \) |
| 73 | \( 1 + (-0.815 - 0.815i)T + 73iT^{2} \) |
| 79 | \( 1 + (-11.8 + 11.8i)T - 79iT^{2} \) |
| 83 | \( 1 + (5.88 - 5.88i)T - 83iT^{2} \) |
| 89 | \( 1 + (-6.87 - 6.87i)T + 89iT^{2} \) |
| 97 | \( 1 - 14.2iT - 97T^{2} \) |
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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
−10.43440321950853861636231165529, −9.002591380368514219653979219512, −8.118255219003298337663506262870, −7.51147781865070204220293568577, −6.60383490241372492169445997949, −5.69670989953110172563523567357, −4.77458292745090341913136426815, −4.07197466690661162691484157542, −2.78875427655260416657904724900, −1.55692517528979913384126221804,
0.55008698974440473537525124069, 2.53806850268489982274008452835, 3.66842101282518572076970701461, 4.54701287391310143452217685762, 5.02328539957857152403755181967, 6.23970443536716255275465190836, 7.21365880283378321114191210031, 7.75486787543166525308457559813, 8.791843383383621047456876607637, 9.904518977405003811728016737170