L(s) = 1 | + (0.707 + 0.707i)2-s + (−0.707 − 0.707i)3-s + 1.00i·4-s + (−1.96 − 1.06i)5-s − 1.00i·6-s + (−1.95 − 1.95i)7-s + (−0.707 + 0.707i)8-s + 1.00i·9-s + (−0.633 − 2.14i)10-s + 0.865i·11-s + (0.707 − 0.707i)12-s + (3.63 + 3.63i)13-s − 2.76i·14-s + (0.633 + 2.14i)15-s − 1.00·16-s + (−2.10 + 2.10i)17-s + ⋯ |
L(s) = 1 | + (0.499 + 0.499i)2-s + (−0.408 − 0.408i)3-s + 0.500i·4-s + (−0.878 − 0.477i)5-s − 0.408i·6-s + (−0.738 − 0.738i)7-s + (−0.250 + 0.250i)8-s + 0.333i·9-s + (−0.200 − 0.678i)10-s + 0.260i·11-s + (0.204 − 0.204i)12-s + (1.00 + 1.00i)13-s − 0.738i·14-s + (0.163 + 0.553i)15-s − 0.250·16-s + (−0.510 + 0.510i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00160 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.00160 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.775314 + 0.774070i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.775314 + 0.774070i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.96 + 1.06i)T \) |
| 31 | \( 1 + (-5.55 - 0.316i)T \) |
good | 7 | \( 1 + (1.95 + 1.95i)T + 7iT^{2} \) |
| 11 | \( 1 - 0.865iT - 11T^{2} \) |
| 13 | \( 1 + (-3.63 - 3.63i)T + 13iT^{2} \) |
| 17 | \( 1 + (2.10 - 2.10i)T - 17iT^{2} \) |
| 19 | \( 1 - 0.565iT - 19T^{2} \) |
| 23 | \( 1 + (-6.50 - 6.50i)T + 23iT^{2} \) |
| 29 | \( 1 + 0.232T + 29T^{2} \) |
| 37 | \( 1 + (2.57 - 2.57i)T - 37iT^{2} \) |
| 41 | \( 1 + 11.3T + 41T^{2} \) |
| 43 | \( 1 + (-4.95 - 4.95i)T + 43iT^{2} \) |
| 47 | \( 1 + (-4.14 - 4.14i)T + 47iT^{2} \) |
| 53 | \( 1 + (-0.513 - 0.513i)T + 53iT^{2} \) |
| 59 | \( 1 - 4.60iT - 59T^{2} \) |
| 61 | \( 1 + 2.89iT - 61T^{2} \) |
| 67 | \( 1 + (1.09 + 1.09i)T + 67iT^{2} \) |
| 71 | \( 1 + 9.81T + 71T^{2} \) |
| 73 | \( 1 + (-0.542 - 0.542i)T + 73iT^{2} \) |
| 79 | \( 1 + 9.17T + 79T^{2} \) |
| 83 | \( 1 + (-0.433 - 0.433i)T + 83iT^{2} \) |
| 89 | \( 1 - 3.35T + 89T^{2} \) |
| 97 | \( 1 + (-6.46 - 6.46i)T + 97iT^{2} \) |
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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
−10.42800962103841552086195890134, −9.209706364301860923573155978159, −8.481595578610099567274075830414, −7.46973311273396122661702158211, −6.86984560787845829056674359893, −6.14346418333021404475495500565, −4.93997630780882887853081357402, −4.10841877628719512039374708669, −3.29254619386066028068714057636, −1.30275806650520869197816783531,
0.51468561376264517327840879836, 2.74764131155387667774105600313, 3.34852043907203155201066731143, 4.41096403105670875937638590000, 5.40551769620031493832116205522, 6.30785029580873214020045968555, 7.03141393249376891531699484604, 8.456074489043211931293077064055, 8.995657522264670303418099125155, 10.26094027934096630781674531613