L(s) = 1 | + (1.41 − 0.0554i)2-s + (−0.488 − 0.488i)3-s + (1.99 − 0.156i)4-s + (−0.717 − 0.663i)6-s − 4.71i·7-s + (2.80 − 0.331i)8-s − 2.52i·9-s + (−3.91 + 3.91i)11-s + (−1.05 − 0.897i)12-s + (−0.0878 − 0.0878i)13-s + (−0.261 − 6.66i)14-s + (3.95 − 0.624i)16-s + 4.67·17-s + (−0.139 − 3.56i)18-s + (1.81 + 1.81i)19-s + ⋯ |
L(s) = 1 | + (0.999 − 0.0391i)2-s + (−0.282 − 0.282i)3-s + (0.996 − 0.0783i)4-s + (−0.292 − 0.270i)6-s − 1.78i·7-s + (0.993 − 0.117i)8-s − 0.840i·9-s + (−1.17 + 1.17i)11-s + (−0.303 − 0.259i)12-s + (−0.0243 − 0.0243i)13-s + (−0.0698 − 1.78i)14-s + (0.987 − 0.156i)16-s + 1.13·17-s + (−0.0329 − 0.840i)18-s + (0.415 + 0.415i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.522 + 0.852i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.522 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.97265 - 1.10509i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.97265 - 1.10509i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.41 + 0.0554i)T \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (0.488 + 0.488i)T + 3iT^{2} \) |
| 7 | \( 1 + 4.71iT - 7T^{2} \) |
| 11 | \( 1 + (3.91 - 3.91i)T - 11iT^{2} \) |
| 13 | \( 1 + (0.0878 + 0.0878i)T + 13iT^{2} \) |
| 17 | \( 1 - 4.67T + 17T^{2} \) |
| 19 | \( 1 + (-1.81 - 1.81i)T + 19iT^{2} \) |
| 23 | \( 1 - 1.63iT - 23T^{2} \) |
| 29 | \( 1 + (-3.26 - 3.26i)T + 29iT^{2} \) |
| 31 | \( 1 + 2.12T + 31T^{2} \) |
| 37 | \( 1 + (-3.97 + 3.97i)T - 37iT^{2} \) |
| 41 | \( 1 - 8.25iT - 41T^{2} \) |
| 43 | \( 1 + (-2.27 + 2.27i)T - 43iT^{2} \) |
| 47 | \( 1 + 4.06T + 47T^{2} \) |
| 53 | \( 1 + (5.03 - 5.03i)T - 53iT^{2} \) |
| 59 | \( 1 + (5.16 - 5.16i)T - 59iT^{2} \) |
| 61 | \( 1 + (-7.12 - 7.12i)T + 61iT^{2} \) |
| 67 | \( 1 + (7.49 + 7.49i)T + 67iT^{2} \) |
| 71 | \( 1 + 4.54iT - 71T^{2} \) |
| 73 | \( 1 - 8.30iT - 73T^{2} \) |
| 79 | \( 1 - 11.5T + 79T^{2} \) |
| 83 | \( 1 + (1.16 + 1.16i)T + 83iT^{2} \) |
| 89 | \( 1 + 3.24iT - 89T^{2} \) |
| 97 | \( 1 + 13.9T + 97T^{2} \) |
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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
−11.22144237081698602786599112515, −10.33512095926126988810389055626, −9.787943159740122465936600150568, −7.64640473098342257001074061679, −7.41850607550410874989676745040, −6.35031043832860360365417010405, −5.18099954821067766251614559986, −4.18256199538220922390439864635, −3.16037431254161597579064794721, −1.28237290090231208808413980119,
2.34301571494514906660227187418, 3.18095048296227479223226002243, 4.94672352972963696679649504469, 5.47466631411292742930739837316, 6.18099127677852910887213373203, 7.78032701637979917578257010985, 8.416810933250544639899397042103, 9.817371248761831898107691011732, 10.84154366083045637902448438529, 11.50883343296982819400470788530