L(s) = 1 | + (1.67 − 1.08i)2-s + 1.73i·3-s + (1.64 − 3.64i)4-s + 2.23·5-s + (1.87 + 2.90i)6-s + 0.596i·7-s + (−1.19 − 7.91i)8-s − 2.99·9-s + (3.75 − 2.42i)10-s + 9.27i·11-s + (6.31 + 2.84i)12-s − 23.5·13-s + (0.647 + 1.00i)14-s + 3.87i·15-s + (−10.5 − 11.9i)16-s + 3.97·17-s + ⋯ |
L(s) = 1 | + (0.839 − 0.542i)2-s + 0.577i·3-s + (0.410 − 0.911i)4-s + 0.447·5-s + (0.313 + 0.484i)6-s + 0.0852i·7-s + (−0.149 − 0.988i)8-s − 0.333·9-s + (0.375 − 0.242i)10-s + 0.843i·11-s + (0.526 + 0.237i)12-s − 1.80·13-s + (0.0462 + 0.0715i)14-s + 0.258i·15-s + (−0.662 − 0.749i)16-s + 0.233·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 + 0.410i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.911 + 0.410i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.75774 - 0.377883i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.75774 - 0.377883i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.67 + 1.08i)T \) |
| 3 | \( 1 - 1.73iT \) |
| 5 | \( 1 - 2.23T \) |
good | 7 | \( 1 - 0.596iT - 49T^{2} \) |
| 11 | \( 1 - 9.27iT - 121T^{2} \) |
| 13 | \( 1 + 23.5T + 169T^{2} \) |
| 17 | \( 1 - 3.97T + 289T^{2} \) |
| 19 | \( 1 + 7.04iT - 361T^{2} \) |
| 23 | \( 1 - 32.0iT - 529T^{2} \) |
| 29 | \( 1 - 35.6T + 841T^{2} \) |
| 31 | \( 1 + 59.2iT - 961T^{2} \) |
| 37 | \( 1 + 5.38T + 1.36e3T^{2} \) |
| 41 | \( 1 - 40.0T + 1.68e3T^{2} \) |
| 43 | \( 1 - 36.1iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 74.0iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 2.55T + 2.80e3T^{2} \) |
| 59 | \( 1 - 36.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 8.73T + 3.72e3T^{2} \) |
| 67 | \( 1 - 69.7iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 59.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 83.0T + 5.32e3T^{2} \) |
| 79 | \( 1 - 65.8iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 129. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 130.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 93.1T + 9.40e3T^{2} \) |
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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
−14.77884484210894326496296461998, −13.68024113085065703510549624421, −12.48371982152967120897995157645, −11.54011048322057777993934434244, −10.08096040694238366000585343752, −9.538857673506610035847107003068, −7.24603429283610330005194238906, −5.56116431538507416089103578025, −4.43052892972481700979321892961, −2.50579013517607124162787433933,
2.75098712081465627649227835374, 4.89052710190228291963232149969, 6.24660221299313911241017639546, 7.39392029482911805800171134180, 8.682525459835767074132984446258, 10.49860601051796777610229499492, 12.07222522013281140403499057071, 12.69761140035827819782207273276, 14.09345940105196737697623993639, 14.44540313398092451601713020766