L(s) = 1 | − i·2-s − i·3-s − 4-s − 3.06i·5-s − 6-s + (2.37 − 1.16i)7-s + i·8-s − 9-s − 3.06·10-s + (−3.20 − 0.857i)11-s + i·12-s + 0.338·13-s + (−1.16 − 2.37i)14-s − 3.06·15-s + 16-s − 0.314·17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.577i·3-s − 0.5·4-s − 1.36i·5-s − 0.408·6-s + (0.897 − 0.441i)7-s + 0.353i·8-s − 0.333·9-s − 0.968·10-s + (−0.966 − 0.258i)11-s + 0.288i·12-s + 0.0939·13-s + (−0.312 − 0.634i)14-s − 0.790·15-s + 0.250·16-s − 0.0762·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 + 0.195i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.980 + 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.123910 - 1.25797i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.123910 - 1.25797i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + iT \) |
| 7 | \( 1 + (-2.37 + 1.16i)T \) |
| 11 | \( 1 + (3.20 + 0.857i)T \) |
good | 5 | \( 1 + 3.06iT - 5T^{2} \) |
| 13 | \( 1 - 0.338T + 13T^{2} \) |
| 17 | \( 1 + 0.314T + 17T^{2} \) |
| 19 | \( 1 - 6.09T + 19T^{2} \) |
| 23 | \( 1 + 3.37T + 23T^{2} \) |
| 29 | \( 1 - 4.40iT - 29T^{2} \) |
| 31 | \( 1 + 0.722iT - 31T^{2} \) |
| 37 | \( 1 + 7.49T + 37T^{2} \) |
| 41 | \( 1 + 8.09T + 41T^{2} \) |
| 43 | \( 1 - 4.12iT - 43T^{2} \) |
| 47 | \( 1 + 8.77iT - 47T^{2} \) |
| 53 | \( 1 - 13.5T + 53T^{2} \) |
| 59 | \( 1 + 4.67iT - 59T^{2} \) |
| 61 | \( 1 - 13.7T + 61T^{2} \) |
| 67 | \( 1 + 1.73T + 67T^{2} \) |
| 71 | \( 1 - 13.5T + 71T^{2} \) |
| 73 | \( 1 + 1.68T + 73T^{2} \) |
| 79 | \( 1 + 6.86iT - 79T^{2} \) |
| 83 | \( 1 - 7.11T + 83T^{2} \) |
| 89 | \( 1 - 2.28iT - 89T^{2} \) |
| 97 | \( 1 - 5.08iT - 97T^{2} \) |
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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
−10.72786749226537847804999118329, −9.815979001387397936066280049833, −8.629504048448185849863733070639, −8.202379353520187973401278187279, −7.18568104941381162481611001523, −5.39293883867762323214145815797, −5.00832011201916861455422767352, −3.61409378775509266636767316232, −1.96113103110862890074318438223, −0.817673886851969833633707895561,
2.44219417156601664606991768858, 3.66855369414407649838805663333, 5.02256404491294898790131959299, 5.74411230539092523608071747608, 6.96715616189293754964268039426, 7.74486971127468203374746865038, 8.596508848244568804989094974110, 9.829615713321034522817570123337, 10.43435828467254998801985934334, 11.33199782765245536690144413738