L(s) = 1 | + (0.5 + 0.866i)3-s + (−0.499 + 0.866i)9-s + (0.5 + 0.866i)11-s − 1.73i·17-s + 1.73i·19-s + (0.5 + 0.866i)25-s − 0.999·27-s + (−0.499 + 0.866i)33-s + (−1.5 − 0.866i)41-s + (1.5 − 0.866i)43-s + (0.5 − 0.866i)49-s + (1.49 − 0.866i)51-s + (−1.49 + 0.866i)57-s + (−0.5 + 0.866i)59-s + (−1.5 − 0.866i)67-s + ⋯ |
L(s) = 1 | + (0.5 + 0.866i)3-s + (−0.499 + 0.866i)9-s + (0.5 + 0.866i)11-s − 1.73i·17-s + 1.73i·19-s + (0.5 + 0.866i)25-s − 0.999·27-s + (−0.499 + 0.866i)33-s + (−1.5 − 0.866i)41-s + (1.5 − 0.866i)43-s + (0.5 − 0.866i)49-s + (1.49 − 0.866i)51-s + (−1.49 + 0.866i)57-s + (−0.5 + 0.866i)59-s + (−1.5 − 0.866i)67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.220803001\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.220803001\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.5 - 0.866i)T \) |
good | 5 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 7 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + 1.73iT - T^{2} \) |
| 19 | \( 1 - 1.73iT - T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T + T^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
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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
−10.07443358411088344535432897769, −9.337354543190856321723673837020, −8.775020080361638186348592034790, −7.66951300868047803755641760938, −7.05928280792007312886934696096, −5.71830336706417071697583893626, −4.92844384708860131588241994054, −4.01774777797790538428691430007, −3.12486692057215219076220711185, −1.89784221832519258796381481516,
1.16313552105860819673581777060, 2.49722513095698334688082160395, 3.45136144260439441947776489926, 4.55014463992081048831084977891, 5.96271890973023822389230858360, 6.46729429166528441585294980128, 7.36376255195429644191716471930, 8.389269550096887448173989284572, 8.719966620729926208168790030478, 9.660693034100202583981091375850