L(s) = 1 | + (0.925 − 0.925i)5-s − 7-s + (1.72 + 1.72i)11-s + (0.328 − 0.328i)13-s − 2.34i·17-s + (1.77 + 1.77i)19-s + 6.17i·23-s + 3.28i·25-s + (−0.122 − 0.122i)29-s + 1.74i·31-s + (−0.925 + 0.925i)35-s + (−1.68 − 1.68i)37-s + 2.88·41-s + (−2.77 + 2.77i)43-s − 5.92·47-s + ⋯ |
L(s) = 1 | + (0.413 − 0.413i)5-s − 0.377·7-s + (0.519 + 0.519i)11-s + (0.0910 − 0.0910i)13-s − 0.567i·17-s + (0.408 + 0.408i)19-s + 1.28i·23-s + 0.657i·25-s + (−0.0227 − 0.0227i)29-s + 0.313i·31-s + (−0.156 + 0.156i)35-s + (−0.276 − 0.276i)37-s + 0.451·41-s + (−0.422 + 0.422i)43-s − 0.863·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.686 - 0.726i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.686 - 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.842751817\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.842751817\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 5 | \( 1 + (-0.925 + 0.925i)T - 5iT^{2} \) |
| 11 | \( 1 + (-1.72 - 1.72i)T + 11iT^{2} \) |
| 13 | \( 1 + (-0.328 + 0.328i)T - 13iT^{2} \) |
| 17 | \( 1 + 2.34iT - 17T^{2} \) |
| 19 | \( 1 + (-1.77 - 1.77i)T + 19iT^{2} \) |
| 23 | \( 1 - 6.17iT - 23T^{2} \) |
| 29 | \( 1 + (0.122 + 0.122i)T + 29iT^{2} \) |
| 31 | \( 1 - 1.74iT - 31T^{2} \) |
| 37 | \( 1 + (1.68 + 1.68i)T + 37iT^{2} \) |
| 41 | \( 1 - 2.88T + 41T^{2} \) |
| 43 | \( 1 + (2.77 - 2.77i)T - 43iT^{2} \) |
| 47 | \( 1 + 5.92T + 47T^{2} \) |
| 53 | \( 1 + (0.973 - 0.973i)T - 53iT^{2} \) |
| 59 | \( 1 + (-8.33 - 8.33i)T + 59iT^{2} \) |
| 61 | \( 1 + (-4.28 + 4.28i)T - 61iT^{2} \) |
| 67 | \( 1 + (1.78 + 1.78i)T + 67iT^{2} \) |
| 71 | \( 1 - 8.57iT - 71T^{2} \) |
| 73 | \( 1 + 6.41iT - 73T^{2} \) |
| 79 | \( 1 - 5.38iT - 79T^{2} \) |
| 83 | \( 1 + (-3.46 + 3.46i)T - 83iT^{2} \) |
| 89 | \( 1 + 1.51T + 89T^{2} \) |
| 97 | \( 1 - 15.7T + 97T^{2} \) |
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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
−8.670592425554653360285862814509, −7.69544899595259949232672514560, −7.13330361240974570822453279090, −6.30815139057644390777339663108, −5.50805785516027622256327176410, −4.92376638374665101377319056046, −3.87942763385023385127896116137, −3.15984690175265060427268687285, −1.97694876345710371465655397954, −1.08328010169342176431408113220,
0.59147531826747911614526904926, 1.92263327066137276377564439305, 2.84001898867382272307794087917, 3.66118170746216019343678256306, 4.52614188731596816026249406666, 5.47382208522676535975885855792, 6.39586366931403908321719035597, 6.57702868561140702740533945017, 7.59559154866287931920520247117, 8.501341569141830484766133091565