L(s) = 1 | + 2.34i·2-s − i·3-s − 3.51·4-s + 3.84i·5-s + 2.34·6-s − 3.56i·8-s − 9-s − 9.02·10-s + (0.648 + 3.25i)11-s + 3.51i·12-s − 2.56·13-s + 3.84·15-s + 1.33·16-s − 0.436·17-s − 2.34i·18-s + 5.74·19-s + ⋯ |
L(s) = 1 | + 1.66i·2-s − 0.577i·3-s − 1.75·4-s + 1.71i·5-s + 0.958·6-s − 1.25i·8-s − 0.333·9-s − 2.85·10-s + (0.195 + 0.980i)11-s + 1.01i·12-s − 0.710·13-s + 0.992·15-s + 0.333·16-s − 0.105·17-s − 0.553i·18-s + 1.31·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0396 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0396 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8326659688\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8326659688\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + iT \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (-0.648 - 3.25i)T \) |
good | 2 | \( 1 - 2.34iT - 2T^{2} \) |
| 5 | \( 1 - 3.84iT - 5T^{2} \) |
| 13 | \( 1 + 2.56T + 13T^{2} \) |
| 17 | \( 1 + 0.436T + 17T^{2} \) |
| 19 | \( 1 - 5.74T + 19T^{2} \) |
| 23 | \( 1 + 5.71T + 23T^{2} \) |
| 29 | \( 1 + 1.22iT - 29T^{2} \) |
| 31 | \( 1 - 5.39iT - 31T^{2} \) |
| 37 | \( 1 + 9.95T + 37T^{2} \) |
| 41 | \( 1 - 12.3T + 41T^{2} \) |
| 43 | \( 1 - 8.05iT - 43T^{2} \) |
| 47 | \( 1 + 5.43iT - 47T^{2} \) |
| 53 | \( 1 - 6.51T + 53T^{2} \) |
| 59 | \( 1 + 3.88iT - 59T^{2} \) |
| 61 | \( 1 + 4.47T + 61T^{2} \) |
| 67 | \( 1 + 4.08T + 67T^{2} \) |
| 71 | \( 1 + 5.05T + 71T^{2} \) |
| 73 | \( 1 + 3.52T + 73T^{2} \) |
| 79 | \( 1 + 16.6iT - 79T^{2} \) |
| 83 | \( 1 + 4.69T + 83T^{2} \) |
| 89 | \( 1 + 2.88iT - 89T^{2} \) |
| 97 | \( 1 + 11.2iT - 97T^{2} \) |
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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
−9.841297539147775350930245607294, −9.010958185968493427752130616786, −7.81143870986897051613756897515, −7.42920035675969026206575341838, −6.93072379358039479598455712706, −6.25637004867581760313456377213, −5.49311720315632277585012452108, −4.42973032134671793811731108066, −3.21337499660328804243582383065, −2.09551767577177379746274894310,
0.33301295550778353975874003233, 1.34143771013260094433264890804, 2.55124581822358110762171597399, 3.71142546534119488804110129032, 4.29176646530014638903284783686, 5.18378156205515207458847033167, 5.79713482057810206081654394182, 7.57176599577507050278731829738, 8.546688903295579002925563648871, 9.076105040402579055205252208464