L(s) = 1 | + 2-s + 4-s + 1.41i·5-s + 8-s − 9-s + 1.41i·10-s + 16-s − 18-s + 1.41i·20-s − 1.00·25-s − 1.41i·29-s + 32-s − 36-s − 1.41i·37-s + 1.41i·40-s + 1.41i·41-s + ⋯ |
L(s) = 1 | + 2-s + 4-s + 1.41i·5-s + 8-s − 9-s + 1.41i·10-s + 16-s − 18-s + 1.41i·20-s − 1.00·25-s − 1.41i·29-s + 32-s − 36-s − 1.41i·37-s + 1.41i·40-s + 1.41i·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 - 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 - 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.846842273\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.846842273\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 17 | \( 1 \) |
good | 3 | \( 1 + T^{2} \) |
| 5 | \( 1 - 1.41iT - T^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + 1.41iT - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + 1.41iT - T^{2} \) |
| 41 | \( 1 - 1.41iT - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + 1.41iT - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + 1.41iT - T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 - 1.41iT - T^{2} \) |
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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
−10.37782133896535897058727791420, −9.473336792913580224820822284835, −8.123104844868391067328305122397, −7.46786936584930805246552501014, −6.44387746314973525475619203676, −6.06216260690761447288166444172, −4.98648292657619979258461325750, −3.79083432070679022376672819701, −2.98841318741524794401284343643, −2.20994499824788278960940488802,
1.41958524975628106160777037168, 2.78962838836108292014508091453, 3.86530813933462331101078733180, 4.91609366761207145121749398379, 5.37658945327439208851325846734, 6.28213338680234465645477220488, 7.32600854090312147157857780631, 8.372049316369935133831786348577, 8.847343217806489063264158348341, 9.959446504008254711568932284596