L(s) = 1 | − 1.41i·5-s + 24·13-s − 32.5i·17-s + 23·25-s + 1.41i·29-s + 70·37-s − 69.2i·41-s − 49·49-s + 103. i·53-s − 22·61-s − 33.9i·65-s − 96·73-s − 46·85-s + 168. i·89-s − 144·97-s + ⋯ |
L(s) = 1 | − 0.282i·5-s + 1.84·13-s − 1.91i·17-s + 0.920·25-s + 0.0487i·29-s + 1.89·37-s − 1.69i·41-s − 0.999·49-s + 1.94i·53-s − 0.360·61-s − 0.522i·65-s − 1.31·73-s − 0.541·85-s + 1.89i·89-s − 1.48·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.61249 - 0.512509i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.61249 - 0.512509i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 1.41iT - 25T^{2} \) |
| 7 | \( 1 + 49T^{2} \) |
| 11 | \( 1 - 121T^{2} \) |
| 13 | \( 1 - 24T + 169T^{2} \) |
| 17 | \( 1 + 32.5iT - 289T^{2} \) |
| 19 | \( 1 + 361T^{2} \) |
| 23 | \( 1 - 529T^{2} \) |
| 29 | \( 1 - 1.41iT - 841T^{2} \) |
| 31 | \( 1 + 961T^{2} \) |
| 37 | \( 1 - 70T + 1.36e3T^{2} \) |
| 41 | \( 1 + 69.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 1.84e3T^{2} \) |
| 47 | \( 1 - 2.20e3T^{2} \) |
| 53 | \( 1 - 103. iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 3.48e3T^{2} \) |
| 61 | \( 1 + 22T + 3.72e3T^{2} \) |
| 67 | \( 1 + 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 + 96T + 5.32e3T^{2} \) |
| 79 | \( 1 + 6.24e3T^{2} \) |
| 83 | \( 1 - 6.88e3T^{2} \) |
| 89 | \( 1 - 168. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 144T + 9.40e3T^{2} \) |
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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
−11.40455288084448529170697657602, −10.73444681161266803411605178172, −9.430786796458750080347435256997, −8.761097516980137156698371596482, −7.64792360165116751020659223077, −6.53352508491681543213215877957, −5.44718075636809267901643654807, −4.24301766966859931064342635227, −2.88628165958439073310052056447, −1.00765691716622985797871660799,
1.44704203259757710934521997889, 3.23990056100399394213424254890, 4.32269421092743956461936916476, 5.92560697305733138129199407580, 6.53885104430615464515509703611, 8.034030674122440330313574295014, 8.647113939972459696625117725755, 9.905915598574959002864075178796, 10.86894351261976665100392060064, 11.42697745112321182878120258018