L(s) = 1 | + 0.549i·2-s + (1.68 − 0.420i)3-s + 1.69·4-s − 3.60·5-s + (0.230 + 0.922i)6-s + (1.38 − 2.25i)7-s + 2.03i·8-s + (2.64 − 1.41i)9-s − 1.98i·10-s − 5.41i·11-s + (2.85 − 0.714i)12-s − 1.72i·13-s + (1.23 + 0.761i)14-s + (−6.06 + 1.51i)15-s + 2.28·16-s + 4.50·17-s + ⋯ |
L(s) = 1 | + 0.388i·2-s + (0.970 − 0.242i)3-s + 0.849·4-s − 1.61·5-s + (0.0942 + 0.376i)6-s + (0.524 − 0.851i)7-s + 0.717i·8-s + (0.882 − 0.471i)9-s − 0.626i·10-s − 1.63i·11-s + (0.823 − 0.206i)12-s − 0.479i·13-s + (0.330 + 0.203i)14-s + (−1.56 + 0.391i)15-s + 0.570·16-s + 1.09·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.953 + 0.301i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.953 + 0.301i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.96429 - 0.303598i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.96429 - 0.303598i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.68 + 0.420i)T \) |
| 7 | \( 1 + (-1.38 + 2.25i)T \) |
| 23 | \( 1 + iT \) |
good | 2 | \( 1 - 0.549iT - 2T^{2} \) |
| 5 | \( 1 + 3.60T + 5T^{2} \) |
| 11 | \( 1 + 5.41iT - 11T^{2} \) |
| 13 | \( 1 + 1.72iT - 13T^{2} \) |
| 17 | \( 1 - 4.50T + 17T^{2} \) |
| 19 | \( 1 - 6.73iT - 19T^{2} \) |
| 29 | \( 1 - 9.24iT - 29T^{2} \) |
| 31 | \( 1 - 0.592iT - 31T^{2} \) |
| 37 | \( 1 + 8.23T + 37T^{2} \) |
| 41 | \( 1 + 6.44T + 41T^{2} \) |
| 43 | \( 1 + 5.11T + 43T^{2} \) |
| 47 | \( 1 - 2.58T + 47T^{2} \) |
| 53 | \( 1 + 1.00iT - 53T^{2} \) |
| 59 | \( 1 - 0.599T + 59T^{2} \) |
| 61 | \( 1 + 6.29iT - 61T^{2} \) |
| 67 | \( 1 + 0.526T + 67T^{2} \) |
| 71 | \( 1 - 1.86iT - 71T^{2} \) |
| 73 | \( 1 - 9.34iT - 73T^{2} \) |
| 79 | \( 1 - 1.25T + 79T^{2} \) |
| 83 | \( 1 - 11.0T + 83T^{2} \) |
| 89 | \( 1 - 3.79T + 89T^{2} \) |
| 97 | \( 1 + 2.14iT - 97T^{2} \) |
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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
−10.88682818265387782650259051887, −10.32407488437964919756637569625, −8.460897088076556661718883312233, −8.180227705183994368510516972990, −7.51343437081255894706771585887, −6.71308913157238956258537735595, −5.28660480590710991743336959312, −3.53712023630967876530615351528, −3.40364245330691468741296896357, −1.26941629108631522256040699647,
1.88852339902024474158946356628, 2.94137452046953305136751489747, 4.03845864518156325880594604110, 4.93860519629252708363718839948, 6.87371909509348809409371938279, 7.50691963976419379866390729259, 8.193518263010570405169659603272, 9.257987716560957961993783776805, 10.14234302420888309201620329716, 11.18434510604130111153635212699