L(s) = 1 | − 2-s + (0.403 + 1.68i)3-s + 4-s + (−0.403 − 1.68i)6-s + (−1.28 − 2.31i)7-s − 8-s + (−2.67 + 1.35i)9-s + 5.34i·11-s + (0.403 + 1.68i)12-s − 3.95·13-s + (1.28 + 2.31i)14-s + 16-s − 7.32i·17-s + (2.67 − 1.35i)18-s − 0.807i·19-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (0.232 + 0.972i)3-s + 0.5·4-s + (−0.164 − 0.687i)6-s + (−0.484 − 0.874i)7-s − 0.353·8-s + (−0.891 + 0.453i)9-s + 1.61i·11-s + (0.116 + 0.486i)12-s − 1.09·13-s + (0.342 + 0.618i)14-s + 0.250·16-s − 1.77i·17-s + (0.630 − 0.320i)18-s − 0.185i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.358 + 0.933i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.358 + 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2250179828\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2250179828\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + (-0.403 - 1.68i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (1.28 + 2.31i)T \) |
good | 11 | \( 1 - 5.34iT - 11T^{2} \) |
| 13 | \( 1 + 3.95T + 13T^{2} \) |
| 17 | \( 1 + 7.32iT - 17T^{2} \) |
| 19 | \( 1 + 0.807iT - 19T^{2} \) |
| 23 | \( 1 + 0.281T + 23T^{2} \) |
| 29 | \( 1 - 0.281iT - 29T^{2} \) |
| 31 | \( 1 + 9.07iT - 31T^{2} \) |
| 37 | \( 1 - 6.06iT - 37T^{2} \) |
| 41 | \( 1 + 6.15T + 41T^{2} \) |
| 43 | \( 1 + 6.34iT - 43T^{2} \) |
| 47 | \( 1 - 5.78iT - 47T^{2} \) |
| 53 | \( 1 + 10.9T + 53T^{2} \) |
| 59 | \( 1 + 4.90T + 59T^{2} \) |
| 61 | \( 1 + 13.2iT - 61T^{2} \) |
| 67 | \( 1 + 6.71iT - 67T^{2} \) |
| 71 | \( 1 + 3.36iT - 71T^{2} \) |
| 73 | \( 1 - 4.98T + 73T^{2} \) |
| 79 | \( 1 - 3.26T + 79T^{2} \) |
| 83 | \( 1 + 1.53iT - 83T^{2} \) |
| 89 | \( 1 + 4.31T + 89T^{2} \) |
| 97 | \( 1 + 15.0T + 97T^{2} \) |
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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
−9.621408553518674862011783098238, −9.345096216765053191969359706945, −7.898230252980519357260451882936, −7.36472325983447955620523808992, −6.53686186708676862548738346437, −5.01807038064436214642296925855, −4.50063142774715091714885974522, −3.20973953109291483302919237340, −2.20628937978825339825996412098, −0.11780701823042280933456622933,
1.51218222279076329315901243464, 2.67176842050924539014452602717, 3.49848737318290668609070782845, 5.44424550300012422159086149317, 6.12678943727492428618426574237, 6.81835731176888339218003744216, 7.912410958791130109225002417713, 8.526062533165831076421945245874, 9.022031959032099531275480637619, 10.09477441490109530564613311594