L(s) = 1 | + 3.89e3i·3-s + 3.49e4i·5-s + (5.78e5 − 5.86e5i)7-s − 1.03e7·9-s + 3.17e7·11-s + 1.31e7i·13-s − 1.36e8·15-s − 6.48e8i·17-s + 1.38e9i·19-s + (2.28e9 + 2.25e9i)21-s − 2.14e9·23-s − 1.22e9·25-s − 2.18e10i·27-s − 1.35e10·29-s + 1.88e9i·31-s + ⋯ |
L(s) = 1 | + 1.78i·3-s + 0.447i·5-s + (0.701 − 0.712i)7-s − 2.17·9-s + 1.62·11-s + 0.209i·13-s − 0.796·15-s − 1.57i·17-s + 1.54i·19-s + (1.26 + 1.25i)21-s − 0.630·23-s − 0.199·25-s − 2.09i·27-s − 0.786·29-s + 0.0686i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.712 + 0.701i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (0.712 + 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{15}{2})\) |
\(\approx\) |
\(0.9573418047\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9573418047\) |
\(L(8)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - 3.49e4iT \) |
| 7 | \( 1 + (-5.78e5 + 5.86e5i)T \) |
good | 3 | \( 1 - 3.89e3iT - 4.78e6T^{2} \) |
| 11 | \( 1 - 3.17e7T + 3.79e14T^{2} \) |
| 13 | \( 1 - 1.31e7iT - 3.93e15T^{2} \) |
| 17 | \( 1 + 6.48e8iT - 1.68e17T^{2} \) |
| 19 | \( 1 - 1.38e9iT - 7.99e17T^{2} \) |
| 23 | \( 1 + 2.14e9T + 1.15e19T^{2} \) |
| 29 | \( 1 + 1.35e10T + 2.97e20T^{2} \) |
| 31 | \( 1 - 1.88e9iT - 7.56e20T^{2} \) |
| 37 | \( 1 - 9.85e10T + 9.01e21T^{2} \) |
| 41 | \( 1 + 2.92e11iT - 3.79e22T^{2} \) |
| 43 | \( 1 + 2.83e11T + 7.38e22T^{2} \) |
| 47 | \( 1 + 7.96e11iT - 2.56e23T^{2} \) |
| 53 | \( 1 + 1.76e12T + 1.37e24T^{2} \) |
| 59 | \( 1 + 2.37e12iT - 6.19e24T^{2} \) |
| 61 | \( 1 + 1.38e12iT - 9.87e24T^{2} \) |
| 67 | \( 1 + 7.50e12T + 3.67e25T^{2} \) |
| 71 | \( 1 + 1.56e13T + 8.27e25T^{2} \) |
| 73 | \( 1 + 9.33e12iT - 1.22e26T^{2} \) |
| 79 | \( 1 + 1.71e13T + 3.68e26T^{2} \) |
| 83 | \( 1 - 2.70e12iT - 7.36e26T^{2} \) |
| 89 | \( 1 + 7.66e13iT - 1.95e27T^{2} \) |
| 97 | \( 1 + 1.58e13iT - 6.52e27T^{2} \) |
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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
−10.28540497629857360169323563905, −9.639038500498643130326815822142, −8.720891286156042730691509280946, −7.40274418018139884973912727011, −6.05345965983773292732852521504, −4.86599844224863607466777616622, −3.99576610974136934603167248769, −3.38711830229734076649109843854, −1.73055599524049541736824233679, −0.17096919903993937079039443193,
1.25314632720691526818037963367, 1.56260606810583465553103541390, 2.76130704162639216500690622997, 4.39446165634382010517635714970, 5.88201485688313463155093954526, 6.46672311560732971062989000061, 7.67223017315927089568807457024, 8.469046561772567876755534201142, 9.250876629047041633942446701586, 11.23438658496863485189690002207