L(s) = 1 | + (−0.707 + 0.707i)2-s + (0.853 + 1.50i)3-s − 1.00i·4-s + (−2.20 − 0.358i)5-s + (−1.66 − 0.461i)6-s + (−1.72 − 1.72i)7-s + (0.707 + 0.707i)8-s + (−1.54 + 2.57i)9-s + (1.81 − 1.30i)10-s − 0.339i·11-s + (1.50 − 0.853i)12-s + (2.77 − 2.77i)13-s + 2.43·14-s + (−1.34 − 3.63i)15-s − 1.00·16-s + (−0.287 + 0.287i)17-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.499i)2-s + (0.492 + 0.870i)3-s − 0.500i·4-s + (−0.987 − 0.160i)5-s + (−0.681 − 0.188i)6-s + (−0.650 − 0.650i)7-s + (0.250 + 0.250i)8-s + (−0.513 + 0.857i)9-s + (0.573 − 0.413i)10-s − 0.102i·11-s + (0.435 − 0.246i)12-s + (0.769 − 0.769i)13-s + 0.650·14-s + (−0.346 − 0.937i)15-s − 0.250·16-s + (−0.0697 + 0.0697i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.122i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 - 0.122i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.992464 + 0.0608655i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.992464 + 0.0608655i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 - 0.707i)T \) |
| 3 | \( 1 + (-0.853 - 1.50i)T \) |
| 5 | \( 1 + (2.20 + 0.358i)T \) |
| 31 | \( 1 - T \) |
good | 7 | \( 1 + (1.72 + 1.72i)T + 7iT^{2} \) |
| 11 | \( 1 + 0.339iT - 11T^{2} \) |
| 13 | \( 1 + (-2.77 + 2.77i)T - 13iT^{2} \) |
| 17 | \( 1 + (0.287 - 0.287i)T - 17iT^{2} \) |
| 19 | \( 1 + 5.47iT - 19T^{2} \) |
| 23 | \( 1 + (-3.99 - 3.99i)T + 23iT^{2} \) |
| 29 | \( 1 - 6.27T + 29T^{2} \) |
| 37 | \( 1 + (-0.324 - 0.324i)T + 37iT^{2} \) |
| 41 | \( 1 + 8.78iT - 41T^{2} \) |
| 43 | \( 1 + (-6.31 + 6.31i)T - 43iT^{2} \) |
| 47 | \( 1 + (-2.27 + 2.27i)T - 47iT^{2} \) |
| 53 | \( 1 + (-7.53 - 7.53i)T + 53iT^{2} \) |
| 59 | \( 1 - 0.628T + 59T^{2} \) |
| 61 | \( 1 + 11.4T + 61T^{2} \) |
| 67 | \( 1 + (-0.339 - 0.339i)T + 67iT^{2} \) |
| 71 | \( 1 - 1.05iT - 71T^{2} \) |
| 73 | \( 1 + (-6.50 + 6.50i)T - 73iT^{2} \) |
| 79 | \( 1 - 3.87iT - 79T^{2} \) |
| 83 | \( 1 + (6.20 + 6.20i)T + 83iT^{2} \) |
| 89 | \( 1 + 13.1T + 89T^{2} \) |
| 97 | \( 1 + (-0.395 - 0.395i)T + 97iT^{2} \) |
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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
−10.06775804054121580396331992911, −8.951891308485641606001114033015, −8.658143193801248315193756010719, −7.58667180138597573939149837737, −7.00041150478912991374592743958, −5.71153632338425824997836439011, −4.69776788225066973656391675068, −3.77749689111856976007340224318, −2.91486562162842413963033552953, −0.64269396492481409334658078981,
1.11709328106142984636917366412, 2.56294877904484499492571047123, 3.34929386111565874683420061583, 4.38306915744071993495330764250, 6.13591015591222430425811669339, 6.75438616006329541185336256757, 7.73663767886073946287099343132, 8.451715427495517139427226029470, 9.000681388247059095716660276794, 9.941316410501796141429391308866