L(s) = 1 | + (−0.707 + 0.707i)3-s + (−1.07 + 1.07i)5-s + (−0.929 + 2.47i)7-s − 1.00i·9-s + (1.12 − 1.12i)11-s + (−0.380 − 0.380i)13-s − 1.51i·15-s + 4.93i·17-s + (−0.00735 + 0.00735i)19-s + (−1.09 − 2.40i)21-s − 1.62·23-s + 2.70i·25-s + (0.707 + 0.707i)27-s + (−3.69 + 3.69i)29-s − 1.21·31-s + ⋯ |
L(s) = 1 | + (−0.408 + 0.408i)3-s + (−0.479 + 0.479i)5-s + (−0.351 + 0.936i)7-s − 0.333i·9-s + (0.338 − 0.338i)11-s + (−0.105 − 0.105i)13-s − 0.391i·15-s + 1.19i·17-s + (−0.00168 + 0.00168i)19-s + (−0.238 − 0.525i)21-s − 0.338·23-s + 0.540i·25-s + (0.136 + 0.136i)27-s + (−0.686 + 0.686i)29-s − 0.217·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.911 + 0.412i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.911 + 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3485084239\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3485084239\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.707 - 0.707i)T \) |
| 7 | \( 1 + (0.929 - 2.47i)T \) |
good | 5 | \( 1 + (1.07 - 1.07i)T - 5iT^{2} \) |
| 11 | \( 1 + (-1.12 + 1.12i)T - 11iT^{2} \) |
| 13 | \( 1 + (0.380 + 0.380i)T + 13iT^{2} \) |
| 17 | \( 1 - 4.93iT - 17T^{2} \) |
| 19 | \( 1 + (0.00735 - 0.00735i)T - 19iT^{2} \) |
| 23 | \( 1 + 1.62T + 23T^{2} \) |
| 29 | \( 1 + (3.69 - 3.69i)T - 29iT^{2} \) |
| 31 | \( 1 + 1.21T + 31T^{2} \) |
| 37 | \( 1 + (6.70 + 6.70i)T + 37iT^{2} \) |
| 41 | \( 1 + 2.08T + 41T^{2} \) |
| 43 | \( 1 + (-7.89 + 7.89i)T - 43iT^{2} \) |
| 47 | \( 1 + 2.09T + 47T^{2} \) |
| 53 | \( 1 + (5.28 + 5.28i)T + 53iT^{2} \) |
| 59 | \( 1 + (9.39 + 9.39i)T + 59iT^{2} \) |
| 61 | \( 1 + (-2.45 - 2.45i)T + 61iT^{2} \) |
| 67 | \( 1 + (4.70 + 4.70i)T + 67iT^{2} \) |
| 71 | \( 1 + 5.16T + 71T^{2} \) |
| 73 | \( 1 - 7.83T + 73T^{2} \) |
| 79 | \( 1 - 4.65iT - 79T^{2} \) |
| 83 | \( 1 + (0.694 - 0.694i)T - 83iT^{2} \) |
| 89 | \( 1 + 16.3T + 89T^{2} \) |
| 97 | \( 1 + 15.8iT - 97T^{2} \) |
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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
−10.12813182511730184460723968140, −9.202942481034434485480587784241, −8.628723881760489271987209718363, −7.61248721129835280394197513109, −6.68326448582102674924454285146, −5.88501005243037013259053564549, −5.20257238906199021842452603567, −3.88202543303075439279832562938, −3.29840211284441792727351984887, −1.90097115006296075934138796706,
0.15497581527284005596997987315, 1.39323746323738568274261647547, 2.91824791264485567769014387948, 4.16674143258801987855593356840, 4.74396007887815506448445205296, 5.91079991357687822007055817888, 6.82254383119254270071591424313, 7.44828696910293085541424943530, 8.165248526968201187331409009171, 9.255899267981452426453936630626