L(s) = 1 | + (0.540 + 0.540i)2-s + (0.707 + 0.707i)3-s − 1.41i·4-s + (−1.03 + 1.98i)5-s + 0.763i·6-s + (2.57 + 0.614i)7-s + (1.84 − 1.84i)8-s + 1.00i·9-s + (−1.63 + 0.510i)10-s − 3.85·11-s + (1.00 − 1.00i)12-s + (−3.66 − 3.66i)13-s + (1.05 + 1.72i)14-s + (−2.13 + 0.668i)15-s − 0.839·16-s + (1.49 − 1.49i)17-s + ⋯ |
L(s) = 1 | + (0.381 + 0.381i)2-s + (0.408 + 0.408i)3-s − 0.708i·4-s + (−0.463 + 0.886i)5-s + 0.311i·6-s + (0.972 + 0.232i)7-s + (0.652 − 0.652i)8-s + 0.333i·9-s + (−0.515 + 0.161i)10-s − 1.16·11-s + (0.289 − 0.289i)12-s + (−1.01 − 1.01i)13-s + (0.282 + 0.460i)14-s + (−0.550 + 0.172i)15-s − 0.209·16-s + (0.361 − 0.361i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.812 - 0.583i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.812 - 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.24914 + 0.402038i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.24914 + 0.402038i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.707 - 0.707i)T \) |
| 5 | \( 1 + (1.03 - 1.98i)T \) |
| 7 | \( 1 + (-2.57 - 0.614i)T \) |
good | 2 | \( 1 + (-0.540 - 0.540i)T + 2iT^{2} \) |
| 11 | \( 1 + 3.85T + 11T^{2} \) |
| 13 | \( 1 + (3.66 + 3.66i)T + 13iT^{2} \) |
| 17 | \( 1 + (-1.49 + 1.49i)T - 17iT^{2} \) |
| 19 | \( 1 - 0.0697T + 19T^{2} \) |
| 23 | \( 1 + (0.534 - 0.534i)T - 23iT^{2} \) |
| 29 | \( 1 - 2.77iT - 29T^{2} \) |
| 31 | \( 1 + 2.39iT - 31T^{2} \) |
| 37 | \( 1 + (-6.18 - 6.18i)T + 37iT^{2} \) |
| 41 | \( 1 - 8.68iT - 41T^{2} \) |
| 43 | \( 1 + (2.77 - 2.77i)T - 43iT^{2} \) |
| 47 | \( 1 + (-5.49 + 5.49i)T - 47iT^{2} \) |
| 53 | \( 1 + (-6.13 + 6.13i)T - 53iT^{2} \) |
| 59 | \( 1 + 6.97T + 59T^{2} \) |
| 61 | \( 1 - 14.3iT - 61T^{2} \) |
| 67 | \( 1 + (-0.416 - 0.416i)T + 67iT^{2} \) |
| 71 | \( 1 + 8.12T + 71T^{2} \) |
| 73 | \( 1 + (-9.55 - 9.55i)T + 73iT^{2} \) |
| 79 | \( 1 + 9.86iT - 79T^{2} \) |
| 83 | \( 1 + (-1.63 - 1.63i)T + 83iT^{2} \) |
| 89 | \( 1 + 5.05T + 89T^{2} \) |
| 97 | \( 1 + (-6.85 + 6.85i)T - 97iT^{2} \) |
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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
−14.22550512591727065934597684106, −13.15614665906510407811310462564, −11.61417334397600777814516906832, −10.54356611686494288994117337407, −9.909525585556248172347520835524, −8.097694927807867396376461256378, −7.31319864194481956553868314226, −5.60915817274846343982356237311, −4.63607480728267832125508198744, −2.72972439489513917433532881784,
2.21015549115096211005793786033, 4.07991046757765507334050106279, 5.10010336689761013924167855500, 7.46834052262121055454367194578, 7.979236857424439912323039164949, 9.091442861830816726274415863150, 10.80646031671678837149610147589, 11.95759074583157083079980706283, 12.50069019940658865073547236293, 13.53049574492770267778626534289