L(s) = 1 | + 4.45i·2-s − 11.8·4-s + 5.08i·7-s − 17.3i·8-s + 58.3·11-s − 21.2i·13-s − 22.6·14-s − 17.8·16-s − 68.8i·17-s + 40.8·19-s + 259. i·22-s + 144. i·23-s + 94.5·26-s − 60.3i·28-s + 220.·29-s + ⋯ |
L(s) = 1 | + 1.57i·2-s − 1.48·4-s + 0.274i·7-s − 0.765i·8-s + 1.59·11-s − 0.452i·13-s − 0.432·14-s − 0.278·16-s − 0.982i·17-s + 0.492·19-s + 2.51i·22-s + 1.30i·23-s + 0.713·26-s − 0.407i·28-s + 1.40·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.066552785\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.066552785\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 - 4.45iT - 8T^{2} \) |
| 7 | \( 1 - 5.08iT - 343T^{2} \) |
| 11 | \( 1 - 58.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 21.2iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 68.8iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 40.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 144. iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 220.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 291.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 260. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 169.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 438. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 255. iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 214. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 331.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 54.9T + 2.26e5T^{2} \) |
| 67 | \( 1 - 758. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 904.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 866. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 206.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 463. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 601.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 229. iT - 9.12e5T^{2} \) |
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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
−10.06516071195851836364949622287, −9.314668480852788417837538947453, −8.556515807850658492927754305236, −7.73661330562891777316936297334, −6.80815312088589572652749128943, −6.24742368448182769348544462212, −5.23199903959049901851841153383, −4.39741285259226458252218332700, −3.02026810687889175972061363407, −1.09589175438936296242091522294,
0.73932677432079617476215513529, 1.67706990240062772774698288754, 2.88112545707199655555854892498, 4.00476771587636696514289442048, 4.50970882978343909782239815263, 6.17781812693932166450819991983, 6.97599002467842863020319271279, 8.487180532724768350777185577569, 9.056191105774503326627644755119, 10.03935421795938782768611426263