| L(s) = 1 | − 1.41i·5-s − 2·7-s + 1.41i·11-s − 1.41i·17-s + (−1 − 4.24i)19-s + 1.41i·23-s + 2.99·25-s − 6·29-s + 2.82i·35-s + 8.48i·37-s − 6·41-s + 4·43-s − 7.07i·47-s − 3·49-s − 6·53-s + ⋯ |
| L(s) = 1 | − 0.632i·5-s − 0.755·7-s + 0.426i·11-s − 0.342i·17-s + (−0.229 − 0.973i)19-s + 0.294i·23-s + 0.599·25-s − 1.11·29-s + 0.478i·35-s + 1.39i·37-s − 0.937·41-s + 0.609·43-s − 1.03i·47-s − 0.428·49-s − 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.927 - 0.374i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.927 - 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (1 + 4.24i)T \) |
| good | 5 | \( 1 + 1.41iT - 5T^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 - 1.41iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 1.41iT - 17T^{2} \) |
| 23 | \( 1 - 1.41iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 8.48iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 7.07iT - 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 4T + 61T^{2} \) |
| 67 | \( 1 - 8.48iT - 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + 10T + 73T^{2} \) |
| 79 | \( 1 - 8.48iT - 79T^{2} \) |
| 83 | \( 1 + 15.5iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 8.48iT - 97T^{2} \) |
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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
−8.601433684672441509933749686453, −7.55710854004184165992271798827, −6.90721777719929407468632365187, −6.14533025279164946267054485543, −5.16211025902678104156867854742, −4.56984495967064686692884362074, −3.51518442895303260884905164114, −2.62789003179984182275392760250, −1.38163295175431056396445053206, 0,
1.66307943256145841616912654843, 2.86170632736332335582529912842, 3.52475650272781103264045855467, 4.40718883959655407034679644922, 5.65065048162430314843615952902, 6.15742851137146798289555343564, 6.94671831811922356897585384104, 7.67127216582801290875691661686, 8.503921129058680073655437423422
