| L(s) = 1 | − 0.208·2-s + 0.469i·3-s − 1.95·4-s − 5-s − 0.0976i·6-s + 2.46i·7-s + 0.823·8-s + 2.77·9-s + 0.208·10-s + (−1.36 − 3.02i)11-s − 0.917i·12-s + 4.03·13-s − 0.513i·14-s − 0.469i·15-s + 3.74·16-s + 0.922i·17-s + ⋯ |
| L(s) = 1 | − 0.147·2-s + 0.270i·3-s − 0.978·4-s − 0.447·5-s − 0.0398i·6-s + 0.932i·7-s + 0.291·8-s + 0.926·9-s + 0.0658·10-s + (−0.412 − 0.911i)11-s − 0.264i·12-s + 1.11·13-s − 0.137i·14-s − 0.121i·15-s + 0.935·16-s + 0.223i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.299 - 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.299 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8569049054\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8569049054\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 11 | \( 1 + (1.36 + 3.02i)T \) |
| 19 | \( 1 + (4.32 + 0.522i)T \) |
| good | 2 | \( 1 + 0.208T + 2T^{2} \) |
| 3 | \( 1 - 0.469iT - 3T^{2} \) |
| 7 | \( 1 - 2.46iT - 7T^{2} \) |
| 13 | \( 1 - 4.03T + 13T^{2} \) |
| 17 | \( 1 - 0.922iT - 17T^{2} \) |
| 23 | \( 1 - 0.0784T + 23T^{2} \) |
| 29 | \( 1 + 2.35T + 29T^{2} \) |
| 31 | \( 1 - 8.68iT - 31T^{2} \) |
| 37 | \( 1 - 8.63iT - 37T^{2} \) |
| 41 | \( 1 + 8.99T + 41T^{2} \) |
| 43 | \( 1 - 4.97iT - 43T^{2} \) |
| 47 | \( 1 - 1.86T + 47T^{2} \) |
| 53 | \( 1 - 2.85iT - 53T^{2} \) |
| 59 | \( 1 - 7.93iT - 59T^{2} \) |
| 61 | \( 1 + 1.44iT - 61T^{2} \) |
| 67 | \( 1 - 12.2iT - 67T^{2} \) |
| 71 | \( 1 - 6.57iT - 71T^{2} \) |
| 73 | \( 1 + 12.7iT - 73T^{2} \) |
| 79 | \( 1 - 2.57T + 79T^{2} \) |
| 83 | \( 1 - 5.78iT - 83T^{2} \) |
| 89 | \( 1 + 7.54iT - 89T^{2} \) |
| 97 | \( 1 - 8.63iT - 97T^{2} \) |
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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
−10.23290219527512134351016988600, −9.111597758877097729013195573125, −8.587771004474273650257270067025, −8.067060442089027812022567169714, −6.76293121173742230401743883213, −5.77664739745764642474668206877, −4.90107828773566257821439260459, −4.00404342891820968274839045510, −3.12692205583596424254323912442, −1.33743053917178420992587015364,
0.47402064389093010132409607073, 1.84844723896950504906430212495, 3.82636191920948614659341399123, 4.13767889420493312888045898794, 5.15812184798775089005404090174, 6.44145333873155549878807386515, 7.38667214276230800720763080794, 7.900137806793412215083104882109, 8.827733986737294060401413773020, 9.731697292105138429811487371535
