| L(s) = 1 | + 1.57·5-s + 1.68i·7-s − 3.33·11-s − 0.177i·13-s + (3.53 − 2.13i)17-s + 2.58i·19-s − 3.40i·23-s − 2.52·25-s − 2.46·29-s + 4.84i·31-s + 2.64i·35-s + 5.33·37-s + 4.79i·41-s + 5.48i·43-s + 5.72·47-s + ⋯ |
| L(s) = 1 | + 0.703·5-s + 0.635i·7-s − 1.00·11-s − 0.0492i·13-s + (0.856 − 0.516i)17-s + 0.593i·19-s − 0.709i·23-s − 0.505·25-s − 0.458·29-s + 0.870i·31-s + 0.447i·35-s + 0.877·37-s + 0.748i·41-s + 0.835i·43-s + 0.834·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.186 - 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.186 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.753910152\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.753910152\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 17 | \( 1 + (-3.53 + 2.13i)T \) |
| good | 5 | \( 1 - 1.57T + 5T^{2} \) |
| 7 | \( 1 - 1.68iT - 7T^{2} \) |
| 11 | \( 1 + 3.33T + 11T^{2} \) |
| 13 | \( 1 + 0.177iT - 13T^{2} \) |
| 19 | \( 1 - 2.58iT - 19T^{2} \) |
| 23 | \( 1 + 3.40iT - 23T^{2} \) |
| 29 | \( 1 + 2.46T + 29T^{2} \) |
| 31 | \( 1 - 4.84iT - 31T^{2} \) |
| 37 | \( 1 - 5.33T + 37T^{2} \) |
| 41 | \( 1 - 4.79iT - 41T^{2} \) |
| 43 | \( 1 - 5.48iT - 43T^{2} \) |
| 47 | \( 1 - 5.72T + 47T^{2} \) |
| 53 | \( 1 + 4.52iT - 53T^{2} \) |
| 59 | \( 1 - 6.32iT - 59T^{2} \) |
| 61 | \( 1 - 6.28T + 61T^{2} \) |
| 67 | \( 1 - 15.8iT - 67T^{2} \) |
| 71 | \( 1 + 13.3iT - 71T^{2} \) |
| 73 | \( 1 - 4.38iT - 73T^{2} \) |
| 79 | \( 1 - 7.47iT - 79T^{2} \) |
| 83 | \( 1 - 6.58iT - 83T^{2} \) |
| 89 | \( 1 - 8.64T + 89T^{2} \) |
| 97 | \( 1 - 13.1iT - 97T^{2} \) |
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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
−8.321646931048643026120176208955, −7.82783239768437649788810824716, −6.97976660706802945772823055173, −6.05662350064436920288326151102, −5.55584221632090948693233997801, −4.96583895553948836807173679527, −3.90197789039182364241416331801, −2.82012310652790719506377314398, −2.30463992585204625219134179849, −1.11256539242164303192998073051,
0.49747514281366070560068280866, 1.74528559870353685691633970370, 2.57951813448569394383410252131, 3.58491451200867830838756848677, 4.34369898528026280514313388983, 5.42635077498333132279199319064, 5.71969720371365066101815327917, 6.66979463423966407540679152278, 7.58330597548753658318539444769, 7.84051833834516994768948032792
