L(s) = 1 | + (0.5 + 1.32i)2-s + (−1.50 + 1.32i)4-s + 1.73i·5-s + (−2.50 − 1.32i)8-s + (−2.29 + 0.866i)10-s − 2.64i·11-s − 3.46i·13-s + (0.500 − 3.96i)16-s − 6.92i·17-s + (−2.29 − 2.59i)20-s + (3.50 − 1.32i)22-s − 5.29i·23-s + 2.00·25-s + (4.58 − 1.73i)26-s + 5·29-s + ⋯ |
L(s) = 1 | + (0.353 + 0.935i)2-s + (−0.750 + 0.661i)4-s + 0.774i·5-s + (−0.883 − 0.467i)8-s + (−0.724 + 0.273i)10-s − 0.797i·11-s − 0.960i·13-s + (0.125 − 0.992i)16-s − 1.68i·17-s + (−0.512 − 0.580i)20-s + (0.746 − 0.282i)22-s − 1.10i·23-s + 0.400·25-s + (0.898 − 0.339i)26-s + 0.928·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00900i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.00900i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.420435181\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.420435181\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 1.32i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 1.73iT - 5T^{2} \) |
| 11 | \( 1 + 2.64iT - 11T^{2} \) |
| 13 | \( 1 + 3.46iT - 13T^{2} \) |
| 17 | \( 1 + 6.92iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 5.29iT - 23T^{2} \) |
| 29 | \( 1 - 5T + 29T^{2} \) |
| 31 | \( 1 + 4.58T + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 3.46iT - 41T^{2} \) |
| 43 | \( 1 - 10.5iT - 43T^{2} \) |
| 47 | \( 1 + 9.16T + 47T^{2} \) |
| 53 | \( 1 - 7T + 53T^{2} \) |
| 59 | \( 1 + 13.7T + 59T^{2} \) |
| 61 | \( 1 + 10.3iT - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 5.29iT - 71T^{2} \) |
| 73 | \( 1 - 6.92iT - 73T^{2} \) |
| 79 | \( 1 - 7.93iT - 79T^{2} \) |
| 83 | \( 1 - 4.58T + 83T^{2} \) |
| 89 | \( 1 + 10.3iT - 89T^{2} \) |
| 97 | \( 1 + 8.66iT - 97T^{2} \) |
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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
−9.119943900097734692907940051091, −8.317358427412522792888374583931, −7.62419334038301129903278692239, −6.82192437596368099096505920300, −6.22360270087637323903923809049, −5.29097906716638331819091112170, −4.56869429086689644367297141216, −3.23702105368299946217537973945, −2.82923585454643871322468816201, −0.50153310686929218928549204957,
1.34506238063386268293541022927, 2.06051032286517125151347340219, 3.47815547980126109079049056812, 4.27263952829200180660262764517, 4.95784582400277468118911239323, 5.85125625688500364109899004072, 6.77652433501914747146967726277, 7.936897286165732904909741822246, 8.835418936020237580367058238428, 9.271815460733842328636138174934