| L(s) = 1 | + i·2-s − i·3-s − 4-s + 0.313i·5-s + 6-s − i·8-s − 9-s − 0.313·10-s + (−0.720 − 3.23i)11-s + i·12-s − 4.42·13-s + 0.313·15-s + 16-s + 3.88·17-s − i·18-s − 1.19·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + 0.140i·5-s + 0.408·6-s − 0.353i·8-s − 0.333·9-s − 0.0992·10-s + (−0.217 − 0.976i)11-s + 0.288i·12-s − 1.22·13-s + 0.0810·15-s + 0.250·16-s + 0.941·17-s − 0.235i·18-s − 0.275·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.597 - 0.801i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.597 - 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7669200624\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7669200624\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 + (0.720 + 3.23i)T \) |
| good | 5 | \( 1 - 0.313iT - 5T^{2} \) |
| 13 | \( 1 + 4.42T + 13T^{2} \) |
| 17 | \( 1 - 3.88T + 17T^{2} \) |
| 19 | \( 1 + 1.19T + 19T^{2} \) |
| 23 | \( 1 + 7.80T + 23T^{2} \) |
| 29 | \( 1 - 6.32iT - 29T^{2} \) |
| 31 | \( 1 + 1.97iT - 31T^{2} \) |
| 37 | \( 1 + 3.89T + 37T^{2} \) |
| 41 | \( 1 - 11.2T + 41T^{2} \) |
| 43 | \( 1 - 4.06iT - 43T^{2} \) |
| 47 | \( 1 - 9.92iT - 47T^{2} \) |
| 53 | \( 1 + 11.7T + 53T^{2} \) |
| 59 | \( 1 - 8.87iT - 59T^{2} \) |
| 61 | \( 1 - 11.6T + 61T^{2} \) |
| 67 | \( 1 + 12.2T + 67T^{2} \) |
| 71 | \( 1 - 10.0T + 71T^{2} \) |
| 73 | \( 1 - 4.85T + 73T^{2} \) |
| 79 | \( 1 - 6.61iT - 79T^{2} \) |
| 83 | \( 1 + 13.4T + 83T^{2} \) |
| 89 | \( 1 + 11.7iT - 89T^{2} \) |
| 97 | \( 1 - 11.7iT - 97T^{2} \) |
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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
−8.741431300487390690498633242286, −7.88028762929175383696788958829, −7.59240531543980517394041458583, −6.66511141163086816474507875403, −5.98138599019845482955712937088, −5.34232558607941650364926572369, −4.43648433725206317418065574142, −3.33314966799326182525574874853, −2.48255782578555748492518825495, −1.08237273767773194520089881019,
0.25821799977293440178422350609, 1.88336335247405623440772722630, 2.63043598115336363557456695547, 3.69166669774458167388715963387, 4.46898004488120168224849889361, 5.08252611071746766776542663246, 5.88859753715127512775530230787, 7.01717366696185447484530252309, 7.81030268700746369221532987223, 8.461043042308341786415381732247
