| L(s) = 1 | + 24.3i·3-s − 49i·7-s − 351.·9-s + 59.4·11-s − 873. i·13-s + 414. i·17-s + 1.39e3·19-s + 1.19e3·21-s − 839. i·23-s − 2.64e3i·27-s + 7.59e3·29-s − 7.01e3·31-s + 1.44e3i·33-s + 2.76e3i·37-s + 2.13e4·39-s + ⋯ |
| L(s) = 1 | + 1.56i·3-s − 0.377i·7-s − 1.44·9-s + 0.148·11-s − 1.43i·13-s + 0.348i·17-s + 0.888·19-s + 0.591·21-s − 0.330i·23-s − 0.698i·27-s + 1.67·29-s − 1.31·31-s + 0.231i·33-s + 0.331i·37-s + 2.24·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.540542763\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.540542763\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 49iT \) |
| good | 3 | \( 1 - 24.3iT - 243T^{2} \) |
| 11 | \( 1 - 59.4T + 1.61e5T^{2} \) |
| 13 | \( 1 + 873. iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 414. iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 1.39e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 839. iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 7.59e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 7.01e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 2.76e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 1.43e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.23e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 1.20e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 1.44e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 1.67e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.38e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.21e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 4.38e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.34e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 3.68e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.14e5iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 5.87e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 3.44e4iT - 8.58e9T^{2} \) |
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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
−10.15198231962897488153687626905, −9.401907730492410873417968109080, −8.470698914590745203930215323172, −7.64213192681902716943853199203, −6.32055055593998777182530235515, −5.29951848228191137416422009894, −4.63080446194080842683241507742, −3.57786279045682139657959361865, −2.89922802960447095821994250981, −1.02710116786401036934814793616,
0.36094524465750312621319858640, 1.53548305587248107312654252028, 2.23773492656395960318658563042, 3.49965971347803651510825385468, 4.95050260797734603092493183224, 5.97023425145933195019099949584, 6.91372494441943630709897605259, 7.27463660978365715868002145505, 8.435513296393550910422842706611, 9.046882443274235774152109187216
