L(s) = 1 | + (0.206 − 0.206i)3-s + (1.36 − 1.77i)5-s + (0.707 + 0.707i)7-s + 2.91i·9-s + 5.35i·11-s + (1.26 + 1.26i)13-s + (−0.0834 − 0.646i)15-s + (4.90 − 4.90i)17-s + 2.01·19-s + 0.291·21-s + (−0.630 + 0.630i)23-s + (−1.26 − 4.83i)25-s + (1.22 + 1.22i)27-s − 3.20i·29-s − 7.10i·31-s + ⋯ |
L(s) = 1 | + (0.119 − 0.119i)3-s + (0.610 − 0.791i)5-s + (0.267 + 0.267i)7-s + 0.971i·9-s + 1.61i·11-s + (0.351 + 0.351i)13-s + (−0.0215 − 0.167i)15-s + (1.18 − 1.18i)17-s + 0.463·19-s + 0.0636·21-s + (−0.131 + 0.131i)23-s + (−0.253 − 0.967i)25-s + (0.234 + 0.234i)27-s − 0.595i·29-s − 1.27i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.77593 + 0.0873659i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.77593 + 0.0873659i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1.36 + 1.77i)T \) |
| 7 | \( 1 + (-0.707 - 0.707i)T \) |
good | 3 | \( 1 + (-0.206 + 0.206i)T - 3iT^{2} \) |
| 11 | \( 1 - 5.35iT - 11T^{2} \) |
| 13 | \( 1 + (-1.26 - 1.26i)T + 13iT^{2} \) |
| 17 | \( 1 + (-4.90 + 4.90i)T - 17iT^{2} \) |
| 19 | \( 1 - 2.01T + 19T^{2} \) |
| 23 | \( 1 + (0.630 - 0.630i)T - 23iT^{2} \) |
| 29 | \( 1 + 3.20iT - 29T^{2} \) |
| 31 | \( 1 + 7.10iT - 31T^{2} \) |
| 37 | \( 1 + (-3.03 + 3.03i)T - 37iT^{2} \) |
| 41 | \( 1 + 1.34T + 41T^{2} \) |
| 43 | \( 1 + (7.06 - 7.06i)T - 43iT^{2} \) |
| 47 | \( 1 + (-6.08 - 6.08i)T + 47iT^{2} \) |
| 53 | \( 1 + (-6.86 - 6.86i)T + 53iT^{2} \) |
| 59 | \( 1 + 12.6T + 59T^{2} \) |
| 61 | \( 1 + 5.70T + 61T^{2} \) |
| 67 | \( 1 + (7.40 + 7.40i)T + 67iT^{2} \) |
| 71 | \( 1 + 9.52iT - 71T^{2} \) |
| 73 | \( 1 + (1.69 + 1.69i)T + 73iT^{2} \) |
| 79 | \( 1 - 9.49T + 79T^{2} \) |
| 83 | \( 1 + (-2.74 + 2.74i)T - 83iT^{2} \) |
| 89 | \( 1 - 12.0iT - 89T^{2} \) |
| 97 | \( 1 + (5.08 - 5.08i)T - 97iT^{2} \) |
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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
−10.67966158051932593499647215670, −9.620089021062791267558393165408, −9.324469556284425119758312070043, −7.914602050590789308285791343938, −7.49063961295449077300771572818, −6.05814772940224182489438089998, −5.08149997083575112193885559575, −4.43311358061937959251692842614, −2.56438418949399801675309418471, −1.52432548164814634376730441678,
1.25717390263027138321404100579, 3.15180812887161491705418269922, 3.63747488676179369904419167071, 5.47161390959953719248969070055, 6.10570909969613522958614668029, 7.03066354326702814233053489472, 8.243980552350501741272867041445, 8.912712577232417861733717741078, 10.12966620397237304711890263902, 10.55528837090189490273947594964