| L(s) = 1 | + 2.23i·2-s − 3.00·4-s − 2.23i·8-s − 4.47i·11-s + 1.73i·13-s − 0.999·16-s − 7.74·17-s + 3.46i·19-s + 10.0·22-s − 4.47i·23-s − 5·25-s − 3.87·26-s − 4.47i·29-s − 1.73i·31-s − 6.70i·32-s + ⋯ |
| L(s) = 1 | + 1.58i·2-s − 1.50·4-s − 0.790i·8-s − 1.34i·11-s + 0.480i·13-s − 0.249·16-s − 1.87·17-s + 0.794i·19-s + 2.13·22-s − 0.932i·23-s − 25-s − 0.759·26-s − 0.830i·29-s − 0.311i·31-s − 1.18i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.654 + 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.654 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2454947405\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2454947405\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 2.23iT - 2T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 + 4.47iT - 11T^{2} \) |
| 13 | \( 1 - 1.73iT - 13T^{2} \) |
| 17 | \( 1 + 7.74T + 17T^{2} \) |
| 19 | \( 1 - 3.46iT - 19T^{2} \) |
| 23 | \( 1 + 4.47iT - 23T^{2} \) |
| 29 | \( 1 + 4.47iT - 29T^{2} \) |
| 31 | \( 1 + 1.73iT - 31T^{2} \) |
| 37 | \( 1 - 5T + 37T^{2} \) |
| 41 | \( 1 + 7.74T + 41T^{2} \) |
| 43 | \( 1 + 7T + 43T^{2} \) |
| 47 | \( 1 + 7.74T + 47T^{2} \) |
| 53 | \( 1 + 4.47iT - 53T^{2} \) |
| 59 | \( 1 - 7.74T + 59T^{2} \) |
| 61 | \( 1 + 8.66iT - 61T^{2} \) |
| 67 | \( 1 + T + 67T^{2} \) |
| 71 | \( 1 - 8.94iT - 71T^{2} \) |
| 73 | \( 1 - 6.92iT - 73T^{2} \) |
| 79 | \( 1 - 11T + 79T^{2} \) |
| 83 | \( 1 - 7.74T + 83T^{2} \) |
| 89 | \( 1 + 15.4T + 89T^{2} \) |
| 97 | \( 1 + 1.73iT - 97T^{2} \) |
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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
−9.165813669317680058367006428678, −8.321052200996870264186430778956, −8.098363366270852477232686379489, −6.76091468600867781463162811670, −6.42416062462811241108957657743, −5.60660937413714552342491363907, −4.63597038455383774019261599012, −3.77611291281668921802120699173, −2.21746838255265038496659161753, −0.096030460006892425022648807939,
1.62016671749628625577025097916, 2.39766806998932388034598561324, 3.46179562049907954120705714476, 4.44878837514606012685335661438, 5.06192291845201827909245424734, 6.54944068419191107695854603579, 7.29516654260928200783055093786, 8.477537100337198488906184936098, 9.294836141103984663809221480173, 9.844628517345921754315145790796
