L(s) = 1 | + 0.593i·3-s − 3.87i·5-s − 2.92i·7-s + 2.64·9-s + 4.95i·11-s − 3.80·13-s + 2.29·15-s + (2.88 − 2.94i)17-s + 1.82·19-s + 1.73·21-s − 8.40i·23-s − 10.0·25-s + 3.35i·27-s + 1.64i·29-s − 7.62i·31-s + ⋯ |
L(s) = 1 | + 0.342i·3-s − 1.73i·5-s − 1.10i·7-s + 0.882·9-s + 1.49i·11-s − 1.05·13-s + 0.593·15-s + (0.699 − 0.714i)17-s + 0.417·19-s + 0.379·21-s − 1.75i·23-s − 2.00·25-s + 0.644i·27-s + 0.305i·29-s − 1.36i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.699 + 0.714i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.699 + 0.714i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.611405055\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.611405055\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 17 | \( 1 + (-2.88 + 2.94i)T \) |
| 59 | \( 1 - T \) |
good | 3 | \( 1 - 0.593iT - 3T^{2} \) |
| 5 | \( 1 + 3.87iT - 5T^{2} \) |
| 7 | \( 1 + 2.92iT - 7T^{2} \) |
| 11 | \( 1 - 4.95iT - 11T^{2} \) |
| 13 | \( 1 + 3.80T + 13T^{2} \) |
| 19 | \( 1 - 1.82T + 19T^{2} \) |
| 23 | \( 1 + 8.40iT - 23T^{2} \) |
| 29 | \( 1 - 1.64iT - 29T^{2} \) |
| 31 | \( 1 + 7.62iT - 31T^{2} \) |
| 37 | \( 1 + 0.0884iT - 37T^{2} \) |
| 41 | \( 1 + 2.48iT - 41T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 47 | \( 1 + 1.02T + 47T^{2} \) |
| 53 | \( 1 - 0.471T + 53T^{2} \) |
| 61 | \( 1 - 6.85iT - 61T^{2} \) |
| 67 | \( 1 + 12.9T + 67T^{2} \) |
| 71 | \( 1 + 8.36iT - 71T^{2} \) |
| 73 | \( 1 - 7.15iT - 73T^{2} \) |
| 79 | \( 1 + 8.40iT - 79T^{2} \) |
| 83 | \( 1 - 2.81T + 83T^{2} \) |
| 89 | \( 1 + 12.6T + 89T^{2} \) |
| 97 | \( 1 + 2.93iT - 97T^{2} \) |
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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
−8.067639449985279363230039609755, −7.37698344491280718803490984131, −7.07777243172986761795838268352, −5.71968279892840905165035848870, −4.76758483493736316836707629093, −4.53918906599465811465540667500, −3.99883570560042612270793511801, −2.44405171533912485718144679787, −1.36819407317292299403366897622, −0.48183729257110371298599808348,
1.43988703640224016448567008024, 2.53235202496230858516152534092, 3.13804305155213003477616073927, 3.85061102691203002203130227064, 5.27842734724771013507956026955, 5.88344854917726497162554112474, 6.49272186406028540472745572024, 7.36891768519928730407984527416, 7.70429088978262985482652074821, 8.648195561545209161809000591961