| L(s) = 1 | + 1.83·5-s + 1.64i·7-s − 3.26·11-s − 3.78·13-s − 6.03·17-s − 2.35i·19-s + (−3.44 + 3.33i)23-s − 1.63·25-s − 7.92i·29-s − 7.60·31-s + 3.01i·35-s − 0.936i·37-s − 8.18i·41-s + 11.8i·43-s − 5.94i·47-s + ⋯ |
| L(s) = 1 | + 0.819·5-s + 0.620i·7-s − 0.983·11-s − 1.05·13-s − 1.46·17-s − 0.540i·19-s + (−0.718 + 0.695i)23-s − 0.327·25-s − 1.47i·29-s − 1.36·31-s + 0.509i·35-s − 0.154i·37-s − 1.27i·41-s + 1.80i·43-s − 0.866i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1656 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 + 0.153i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1656 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.988 + 0.153i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.009911208352\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.009911208352\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 23 | \( 1 + (3.44 - 3.33i)T \) |
| good | 5 | \( 1 - 1.83T + 5T^{2} \) |
| 7 | \( 1 - 1.64iT - 7T^{2} \) |
| 11 | \( 1 + 3.26T + 11T^{2} \) |
| 13 | \( 1 + 3.78T + 13T^{2} \) |
| 17 | \( 1 + 6.03T + 17T^{2} \) |
| 19 | \( 1 + 2.35iT - 19T^{2} \) |
| 29 | \( 1 + 7.92iT - 29T^{2} \) |
| 31 | \( 1 + 7.60T + 31T^{2} \) |
| 37 | \( 1 + 0.936iT - 37T^{2} \) |
| 41 | \( 1 + 8.18iT - 41T^{2} \) |
| 43 | \( 1 - 11.8iT - 43T^{2} \) |
| 47 | \( 1 + 5.94iT - 47T^{2} \) |
| 53 | \( 1 - 1.87T + 53T^{2} \) |
| 59 | \( 1 + 7.09iT - 59T^{2} \) |
| 61 | \( 1 + 2.92iT - 61T^{2} \) |
| 67 | \( 1 - 13.4iT - 67T^{2} \) |
| 71 | \( 1 - 1.06iT - 71T^{2} \) |
| 73 | \( 1 - 13.8T + 73T^{2} \) |
| 79 | \( 1 + 0.0827iT - 79T^{2} \) |
| 83 | \( 1 - 13.4T + 83T^{2} \) |
| 89 | \( 1 + 13.8T + 89T^{2} \) |
| 97 | \( 1 + 4.44iT - 97T^{2} \) |
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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
−9.626485023248813803657110839453, −9.211187778265421004722758691562, −8.199719090592866298888061245534, −7.42796537613961260274221764141, −6.50487809505881461811798576213, −5.62175874829046848890596471726, −5.07317952437680803952257177463, −3.97010703724083471030812444841, −2.42817338940478528001936416973, −2.17142145152401097309028556353,
0.00327626771662123260717537289, 1.83400466665881333646571750172, 2.63087368090133320710826054647, 3.93661824107106110168462472402, 4.89498017528993286562347276274, 5.61326495197324542273167955894, 6.60555313494840950802726384779, 7.30858155301834585695814195243, 8.122930128762939345898683992339, 9.078095074534571721880268286328
