L(s) = 1 | + (−0.258 + 0.965i)3-s + (1.70 + 2.02i)7-s + (−0.866 − 0.499i)9-s + (1.21 + 2.10i)11-s + (0.728 + 0.728i)13-s + (7.33 + 1.96i)17-s + (1.84 − 3.20i)19-s + (−2.39 + 1.12i)21-s + (−0.759 − 2.83i)23-s + (0.707 − 0.707i)27-s − 1.99i·29-s + (6.02 − 3.47i)31-s + (−2.34 + 0.629i)33-s + (−8.22 + 2.20i)37-s + (−0.892 + 0.515i)39-s + ⋯ |
L(s) = 1 | + (−0.149 + 0.557i)3-s + (0.645 + 0.763i)7-s + (−0.288 − 0.166i)9-s + (0.366 + 0.634i)11-s + (0.202 + 0.202i)13-s + (1.77 + 0.476i)17-s + (0.424 − 0.734i)19-s + (−0.522 + 0.245i)21-s + (−0.158 − 0.591i)23-s + (0.136 − 0.136i)27-s − 0.369i·29-s + (1.08 − 0.624i)31-s + (−0.408 + 0.109i)33-s + (−1.35 + 0.362i)37-s + (−0.142 + 0.0825i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.237 - 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.237 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.941747514\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.941747514\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.258 - 0.965i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-1.70 - 2.02i)T \) |
good | 11 | \( 1 + (-1.21 - 2.10i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.728 - 0.728i)T + 13iT^{2} \) |
| 17 | \( 1 + (-7.33 - 1.96i)T + (14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (-1.84 + 3.20i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.759 + 2.83i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + 1.99iT - 29T^{2} \) |
| 31 | \( 1 + (-6.02 + 3.47i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (8.22 - 2.20i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 - 1.08iT - 41T^{2} \) |
| 43 | \( 1 + (5.91 - 5.91i)T - 43iT^{2} \) |
| 47 | \( 1 + (-1.28 - 4.81i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (-6.56 - 1.76i)T + (45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (1.39 + 2.41i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.217 - 0.125i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.08 + 7.79i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 - 14.5T + 71T^{2} \) |
| 73 | \( 1 + (1.73 - 6.46i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-9.54 - 5.51i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (9.90 + 9.90i)T + 83iT^{2} \) |
| 89 | \( 1 + (5.81 - 10.0i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (10.8 - 10.8i)T - 97iT^{2} \) |
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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
−9.441731078421571673493001032815, −8.414735106205550626097023094303, −7.960389184467024273638956933412, −6.85631412517773968808022812382, −5.99256230776126669428912243475, −5.20173557620940504949903923454, −4.53300935278976544521443180872, −3.52430316818118597116573419413, −2.48676448581843079769019244798, −1.25236972968354954005690145092,
0.838027017626611906849398369426, 1.65389361156943065773847690405, 3.19426904563350972519061257799, 3.83460389264887555271359612586, 5.20230958230678839047548546673, 5.61444468333028021632426836756, 6.76173335043039504801410204370, 7.38434227774294651924613479848, 8.117756507915487494089527928150, 8.685835539743582792638858635057