L(s) = 1 | + (0.5 + 0.866i)2-s + (0.707 − 1.22i)3-s + (−0.499 + 0.866i)4-s + 1.41·6-s − 0.999·8-s + (−0.499 − 0.866i)9-s + (0.707 + 1.22i)12-s + (−0.5 − 0.866i)16-s + (−0.707 + 1.22i)17-s + (0.5 − 0.866i)18-s + (−0.707 − 1.22i)19-s + (−0.707 + 1.22i)24-s + (−0.5 + 0.866i)25-s + (0.499 − 0.866i)32-s − 1.41·34-s + ⋯ |
L(s) = 1 | + (0.5 + 0.866i)2-s + (0.707 − 1.22i)3-s + (−0.499 + 0.866i)4-s + 1.41·6-s − 0.999·8-s + (−0.499 − 0.866i)9-s + (0.707 + 1.22i)12-s + (−0.5 − 0.866i)16-s + (−0.707 + 1.22i)17-s + (0.5 − 0.866i)18-s + (−0.707 − 1.22i)19-s + (−0.707 + 1.22i)24-s + (−0.5 + 0.866i)25-s + (0.499 − 0.866i)32-s − 1.41·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.166623517\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.166623517\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 5 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + 1.41T + T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - 1.41T + T^{2} \) |
| 89 | \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 - 1.41T + T^{2} \) |
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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
−11.99749138160849824781851778020, −10.82177021404789126881442254844, −9.226273715847906902359250565934, −8.507146198468054036463102140639, −7.75105913992276727584293806689, −6.85578042785168392983892937457, −6.20815541429223100426914943401, −4.81185762247012590249160734986, −3.48796433458912739743561166380, −2.13228944372364409470001681880,
2.28883595509610354560748598200, 3.46186598769586487547194133605, 4.31058511544820050170153889786, 5.19064242793329841487128014132, 6.50684546958193927497430349016, 8.217643887931389091092843722523, 9.055542959058167545682038208745, 9.873265699529936988052333387250, 10.41476363770107547413858918359, 11.39093439597806997439417601180