L(s) = 1 | + 5-s + (−1 − i)13-s + (1 − i)17-s + 25-s − 2i·29-s + (−1 + i)37-s + 2·41-s + i·49-s + (1 + i)53-s + (−1 − i)65-s + (−1 − i)73-s + (1 − i)85-s + 2i·89-s + (−1 + i)97-s + 2i·109-s + ⋯ |
L(s) = 1 | + 5-s + (−1 − i)13-s + (1 − i)17-s + 25-s − 2i·29-s + (−1 + i)37-s + 2·41-s + i·49-s + (1 + i)53-s + (−1 − i)65-s + (−1 − i)73-s + (1 − i)85-s + 2i·89-s + (−1 + i)97-s + 2i·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.453638547\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.453638547\) |
\(L(1)\) |
|
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 | \( 1 - T \) |
good | 7 | \( 1 - iT^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + (1 + i)T + iT^{2} \) |
| 17 | \( 1 + (-1 + i)T - iT^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + iT^{2} \) |
| 29 | \( 1 + 2iT - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + (1 - i)T - iT^{2} \) |
| 41 | \( 1 - 2T + T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 - iT^{2} \) |
| 53 | \( 1 + (-1 - i)T + iT^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (1 + i)T + iT^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 - 2iT - T^{2} \) |
| 97 | \( 1 + (1 - i)T - iT^{2} \) |
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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
−9.122985483421329651393304358192, −7.947055717383514059880694882193, −7.53179085325822748549171207671, −6.53701648709589852388108456791, −5.70793900477979294766613176962, −5.21468584215457646832547000997, −4.27348818929452709643991685152, −2.93801762967853872878877683254, −2.43350896458390204821321676454, −1.00589148276618140205811909963,
1.46392526469955693743567413879, 2.28211909805043134554833880301, 3.36024568570201550238938503975, 4.37805761102425759304775548008, 5.33107753825434546417942632052, 5.82704323853452741440798708245, 6.87958905364385779778160339889, 7.28399161595257855237613723379, 8.451572014982720513571784719359, 9.054808555652830358511866688992