L(s) = 1 | + (−0.707 − 0.707i)5-s − 1.41i·17-s + (−1 + i)19-s − 1.41i·23-s + 1.00i·25-s − 2i·31-s − 1.41·47-s + 49-s + (−1 − i)61-s + (−1.41 − 1.41i)83-s + (−1.00 + 1.00i)85-s + 1.41·95-s + (1.41 − 1.41i)107-s + (−1 − i)109-s + 1.41i·113-s + ⋯ |
L(s) = 1 | + (−0.707 − 0.707i)5-s − 1.41i·17-s + (−1 + i)19-s − 1.41i·23-s + 1.00i·25-s − 2i·31-s − 1.41·47-s + 49-s + (−1 − i)61-s + (−1.41 − 1.41i)83-s + (−1.00 + 1.00i)85-s + 1.41·95-s + (1.41 − 1.41i)107-s + (−1 − i)109-s + 1.41i·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7599698121\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7599698121\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.707 + 0.707i)T \) |
good | 7 | \( 1 - T^{2} \) |
| 11 | \( 1 + iT^{2} \) |
| 13 | \( 1 - iT^{2} \) |
| 17 | \( 1 + 1.41iT - T^{2} \) |
| 19 | \( 1 + (1 - i)T - iT^{2} \) |
| 23 | \( 1 + 1.41iT - T^{2} \) |
| 29 | \( 1 + iT^{2} \) |
| 31 | \( 1 + 2iT - T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + 1.41T + T^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 + iT^{2} \) |
| 61 | \( 1 + (1 + i)T + iT^{2} \) |
| 67 | \( 1 + iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + (1.41 + 1.41i)T + iT^{2} \) |
| 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
−8.626326979959014193365358513361, −8.029439484359320539245761121457, −7.35755499035991571675236450104, −6.44092920164108307002060176671, −5.61918525179796089329292847755, −4.62628912855151761824673720445, −4.17916156963403391562130161653, −3.07560481132494690530229411026, −1.97009753626015196969253287083, −0.47495247243934691030232531325,
1.57985382659527447613009622686, 2.79637384413042231503105127177, 3.61863959614059883204302410386, 4.36033224151136259899205857334, 5.34536198699638453747668737923, 6.33991471649824408420453980283, 6.90102418570606520036168071546, 7.67002808866559835394140244406, 8.434098737880985655711942347351, 9.009205017016937245064315289019