L(s) = 1 | + i·7-s + i·9-s + (−1 + i)11-s − i·25-s + (1 + i)29-s + (1 − i)37-s + (1 − i)43-s − 49-s + (−1 + i)53-s − 63-s + (1 + i)67-s − 2i·71-s + (−1 − i)77-s − 81-s + (−1 − i)99-s + ⋯ |
L(s) = 1 | + i·7-s + i·9-s + (−1 + i)11-s − i·25-s + (1 + i)29-s + (1 − i)37-s + (1 − i)43-s − 49-s + (−1 + i)53-s − 63-s + (1 + i)67-s − 2i·71-s + (−1 − i)77-s − 81-s + (−1 − i)99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9100673006\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9100673006\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - iT \) |
good | 3 | \( 1 - iT^{2} \) |
| 5 | \( 1 + iT^{2} \) |
| 11 | \( 1 + (1 - i)T - iT^{2} \) |
| 13 | \( 1 - iT^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - iT^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + (-1 - i)T + iT^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (-1 + i)T - iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + (-1 + i)T - iT^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + (1 - i)T - iT^{2} \) |
| 59 | \( 1 + iT^{2} \) |
| 61 | \( 1 - iT^{2} \) |
| 67 | \( 1 + (-1 - i)T + iT^{2} \) |
| 71 | \( 1 + 2iT - T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−10.49253034841679419901116498745, −9.695331706325934316099428768350, −8.738654398672561236005124218045, −7.967077029167392279358637634251, −7.22197830387360693709115091764, −6.03961790857930706849123714051, −5.16488297306096516280964501904, −4.46458056243964553787634006158, −2.79379861515395458518021952397, −2.07047102946973997264344666576,
0.937534127854486326249396627624, 2.84596026137875311780091498977, 3.73755484378744644691631457286, 4.79108598334720651267298323880, 5.94016264065495036317911344557, 6.68072229107713202942539774618, 7.72561037113203031657624825015, 8.352034758189487661664143514898, 9.478963582900522041203318423557, 10.10255405844579752430599172626