| L(s) = 1 | + (4.41 − 26.6i)3-s + 127.·5-s − 467.·7-s + (−689. − 235. i)9-s − 554.·11-s + 357. i·13-s + (561. − 3.38e3i)15-s − 4.49e3i·17-s − 3.24e3i·19-s + (−2.06e3 + 1.24e4i)21-s − 1.74e4i·23-s + 511.·25-s + (−9.31e3 + 1.73e4i)27-s + 1.74e4·29-s − 8.03e3·31-s + ⋯ |
| L(s) = 1 | + (0.163 − 0.986i)3-s + 1.01·5-s − 1.36·7-s + (−0.946 − 0.322i)9-s − 0.416·11-s + 0.162i·13-s + (0.166 − 1.00i)15-s − 0.915i·17-s − 0.472i·19-s + (−0.223 + 1.34i)21-s − 1.43i·23-s + 0.0327·25-s + (−0.473 + 0.880i)27-s + 0.714·29-s − 0.269·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.163 - 0.986i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.163 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.4647486104\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4647486104\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-4.41 + 26.6i)T \) |
| good | 5 | \( 1 - 127.T + 1.56e4T^{2} \) |
| 7 | \( 1 + 467.T + 1.17e5T^{2} \) |
| 11 | \( 1 + 554.T + 1.77e6T^{2} \) |
| 13 | \( 1 - 357. iT - 4.82e6T^{2} \) |
| 17 | \( 1 + 4.49e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 3.24e3iT - 4.70e7T^{2} \) |
| 23 | \( 1 + 1.74e4iT - 1.48e8T^{2} \) |
| 29 | \( 1 - 1.74e4T + 5.94e8T^{2} \) |
| 31 | \( 1 + 8.03e3T + 8.87e8T^{2} \) |
| 37 | \( 1 - 7.94e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 - 6.18e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 4.54e4iT - 6.32e9T^{2} \) |
| 47 | \( 1 - 1.59e5iT - 1.07e10T^{2} \) |
| 53 | \( 1 - 1.90e5T + 2.21e10T^{2} \) |
| 59 | \( 1 + 3.06e5T + 4.21e10T^{2} \) |
| 61 | \( 1 + 2.51e5iT - 5.15e10T^{2} \) |
| 67 | \( 1 - 2.02e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 + 3.62e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 3.49e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + 7.18e5T + 2.43e11T^{2} \) |
| 83 | \( 1 - 1.38e5T + 3.26e11T^{2} \) |
| 89 | \( 1 - 1.22e6iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 1.15e6T + 8.32e11T^{2} \) |
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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
−10.35742705139337032237107474498, −9.566661225735811907963948174480, −8.804310341703265590968741145015, −7.61960863841668704629011829491, −6.48160776913147375967394735883, −6.23078916512724441590511642418, −4.87653734445411943330537670200, −3.03389736316164954936714612712, −2.45679572846379824932870091492, −1.04312006997351556909285542518,
0.10437270713263148148757655563, 1.98634057355883693160535028966, 3.14910692376095903566513908645, 3.99973174634618858058168892198, 5.58779170424919875797179648928, 5.88769620691662166563839883833, 7.25856952422432638278549070936, 8.634964572799056154988899656283, 9.391148862553978501918757621971, 10.15397640563834200290238781932
