| L(s) = 1 | − 1.41i·2-s + i·3-s − 1.00·4-s + (0.707 + 0.707i)5-s + 1.41·6-s − 9-s + (1.00 − 1.00i)10-s − 1.41·11-s − 1.00i·12-s − i·13-s + (−0.707 + 0.707i)15-s − 0.999·16-s + 1.41i·18-s + (−0.707 − 0.707i)20-s + 2.00i·22-s + ⋯ |
| L(s) = 1 | − 1.41i·2-s + i·3-s − 1.00·4-s + (0.707 + 0.707i)5-s + 1.41·6-s − 9-s + (1.00 − 1.00i)10-s − 1.41·11-s − 1.00i·12-s − i·13-s + (−0.707 + 0.707i)15-s − 0.999·16-s + 1.41i·18-s + (−0.707 − 0.707i)20-s + 2.00i·22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7046978936\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7046978936\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - iT \) |
| 5 | \( 1 + (-0.707 - 0.707i)T \) |
| 13 | \( 1 + iT \) |
| good | 2 | \( 1 + 1.41iT - T^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( 1 + 1.41T + T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 - 1.41T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - 1.41iT - T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + 1.41T + T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 - 1.41T + T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + 1.41iT - T^{2} \) |
| 89 | \( 1 - 1.41T + T^{2} \) |
| 97 | \( 1 + T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.55625969666550831010351960632, −11.18125484904754212596057441347, −10.67969344315519172032264834362, −10.05477952043037237846160495428, −9.257280885009195301406715077245, −7.80245856384292925587611057570, −6.00811072252381319557068017303, −4.84633011308720520843676092457, −3.29967813781756604352129154086, −2.52383809435635533548851648622,
2.23288252098447639708228577616, 4.88969764023033145209072299567, 5.76768944560953259787734307338, 6.67272123123419501087238197541, 7.70808450273608444746121703780, 8.469663218549072951032124486086, 9.440866714430305903006383467153, 10.99106231624300302462606624737, 12.25469729002513209592688854700, 13.21504086688074319830768585323
