L(s) = 1 | + (0.707 + 0.707i)2-s + (0.707 + 0.707i)3-s + 1.00i·4-s + (0.0490 + 2.23i)5-s + 1.00i·6-s + (3.32 + 3.32i)7-s + (−0.707 + 0.707i)8-s + 1.00i·9-s + (−1.54 + 1.61i)10-s + 4.17i·11-s + (−0.707 + 0.707i)12-s + (−2.60 − 2.60i)13-s + 4.70i·14-s + (−1.54 + 1.61i)15-s − 1.00·16-s + (4.75 − 4.75i)17-s + ⋯ |
L(s) = 1 | + (0.499 + 0.499i)2-s + (0.408 + 0.408i)3-s + 0.500i·4-s + (0.0219 + 0.999i)5-s + 0.408i·6-s + (1.25 + 1.25i)7-s + (−0.250 + 0.250i)8-s + 0.333i·9-s + (−0.488 + 0.510i)10-s + 1.25i·11-s + (−0.204 + 0.204i)12-s + (−0.722 − 0.722i)13-s + 1.25i·14-s + (−0.399 + 0.417i)15-s − 0.250·16-s + (1.15 − 1.15i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.686 - 0.727i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.686 - 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.03168 + 2.39143i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.03168 + 2.39143i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.707 - 0.707i)T \) |
| 5 | \( 1 + (-0.0490 - 2.23i)T \) |
| 31 | \( 1 + (1.00 + 5.47i)T \) |
good | 7 | \( 1 + (-3.32 - 3.32i)T + 7iT^{2} \) |
| 11 | \( 1 - 4.17iT - 11T^{2} \) |
| 13 | \( 1 + (2.60 + 2.60i)T + 13iT^{2} \) |
| 17 | \( 1 + (-4.75 + 4.75i)T - 17iT^{2} \) |
| 19 | \( 1 + 7.18iT - 19T^{2} \) |
| 23 | \( 1 + (4.10 + 4.10i)T + 23iT^{2} \) |
| 29 | \( 1 - 6.16T + 29T^{2} \) |
| 37 | \( 1 + (3.18 - 3.18i)T - 37iT^{2} \) |
| 41 | \( 1 - 8.63T + 41T^{2} \) |
| 43 | \( 1 + (-0.501 - 0.501i)T + 43iT^{2} \) |
| 47 | \( 1 + (-0.354 - 0.354i)T + 47iT^{2} \) |
| 53 | \( 1 + (-5.87 - 5.87i)T + 53iT^{2} \) |
| 59 | \( 1 - 2.54iT - 59T^{2} \) |
| 61 | \( 1 + 5.63iT - 61T^{2} \) |
| 67 | \( 1 + (-7.48 - 7.48i)T + 67iT^{2} \) |
| 71 | \( 1 + 13.4T + 71T^{2} \) |
| 73 | \( 1 + (3.37 + 3.37i)T + 73iT^{2} \) |
| 79 | \( 1 + 9.67T + 79T^{2} \) |
| 83 | \( 1 + (-6.43 - 6.43i)T + 83iT^{2} \) |
| 89 | \( 1 + 3.82T + 89T^{2} \) |
| 97 | \( 1 + (4.24 + 4.24i)T + 97iT^{2} \) |
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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
−10.25851999025292066388201961895, −9.549682568679438989618822904433, −8.574797784612879497086077679392, −7.65513298395185107520680038936, −7.20249260042445624136949884106, −5.91395425200017818802254256673, −5.02570314912398286653118210858, −4.43569686144115983165302139007, −2.74611514302912504060358377721, −2.43822130497354683695809807877,
1.11407187631147328940290541141, 1.77395834958850873026502041975, 3.60952558554813902986027604300, 4.15803852414740462426704957990, 5.27448630016547314245247180532, 6.06045056455856913496825504664, 7.49292058390929652856965763304, 8.079583395038236303216131556207, 8.753649314250923751914838134904, 10.00381996109247843669883297624