L(s) = 1 | + (2.22 + 2.22i)2-s + 5.89i·4-s + (−2.67 − 4.22i)5-s + (−1.44 − 1.44i)7-s + (−4.22 + 4.22i)8-s + (3.44 − 15.3i)10-s + 3.34·11-s + (−10.4 + 10.4i)13-s − 6.44i·14-s + 4.79·16-s + (2.65 + 2.65i)17-s − 20.6i·19-s + (24.9 − 15.7i)20-s + (7.44 + 7.44i)22-s + (−16.4 + 16.4i)23-s + ⋯ |
L(s) = 1 | + (1.11 + 1.11i)2-s + 1.47i·4-s + (−0.534 − 0.844i)5-s + (−0.207 − 0.207i)7-s + (−0.528 + 0.528i)8-s + (0.344 − 1.53i)10-s + 0.304·11-s + (−0.803 + 0.803i)13-s − 0.460i·14-s + 0.299·16-s + (0.155 + 0.155i)17-s − 1.08i·19-s + (1.24 − 0.788i)20-s + (0.338 + 0.338i)22-s + (−0.715 + 0.715i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.437 - 0.899i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.437 - 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.41486 + 0.884982i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.41486 + 0.884982i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (2.67 + 4.22i)T \) |
good | 2 | \( 1 + (-2.22 - 2.22i)T + 4iT^{2} \) |
| 7 | \( 1 + (1.44 + 1.44i)T + 49iT^{2} \) |
| 11 | \( 1 - 3.34T + 121T^{2} \) |
| 13 | \( 1 + (10.4 - 10.4i)T - 169iT^{2} \) |
| 17 | \( 1 + (-2.65 - 2.65i)T + 289iT^{2} \) |
| 19 | \( 1 + 20.6iT - 361T^{2} \) |
| 23 | \( 1 + (16.4 - 16.4i)T - 529iT^{2} \) |
| 29 | \( 1 + 0.853iT - 841T^{2} \) |
| 31 | \( 1 + 18.6T + 961T^{2} \) |
| 37 | \( 1 + (-38.0 - 38.0i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 - 28.6T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-22.4 + 22.4i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (19.7 + 19.7i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (28.6 - 28.6i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 + 111. iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 94.0T + 3.72e3T^{2} \) |
| 67 | \( 1 + (54.8 + 54.8i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 - 68T + 5.04e3T^{2} \) |
| 73 | \( 1 + (39.7 - 39.7i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 - 24.4iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-21.1 + 21.1i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 - 94.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-14.5 - 14.5i)T + 9.40e3iT^{2} \) |
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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
−15.74480818719965891638636147491, −14.73415193297376754579948161406, −13.63620005416769260220052926819, −12.67573348063870124180431544918, −11.61319135221186312571730352533, −9.454685288103260934430922298558, −7.906551238530555087140884762061, −6.74213050607634281568134061313, −5.15347458764630957845446569214, −4.00404658036720457325105214560,
2.70560374470531297517706844904, 4.09524338616557848750338902898, 5.87556272708633580394221834634, 7.72991959115636578419192481699, 9.932390179683085489899533334420, 10.92185659550895205759178113033, 12.03210438982907241744861421975, 12.79292057528765604776159924703, 14.32024037263950906768715819308, 14.78777163689253696853119514978