| L(s) = 1 | − 10i·3-s + 7i·7-s − 73·9-s − 40·11-s − 12i·13-s + 58i·17-s − 26·19-s + 70·21-s − 64i·23-s + 460i·27-s + 62·29-s + 252·31-s + 400i·33-s − 26i·37-s − 120·39-s + ⋯ |
| L(s) = 1 | − 1.92i·3-s + 0.377i·7-s − 2.70·9-s − 1.09·11-s − 0.256i·13-s + 0.827i·17-s − 0.313·19-s + 0.727·21-s − 0.580i·23-s + 3.27i·27-s + 0.397·29-s + 1.46·31-s + 2.11i·33-s − 0.115i·37-s − 0.492·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6967658244\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6967658244\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 7iT \) |
| good | 3 | \( 1 + 10iT - 27T^{2} \) |
| 11 | \( 1 + 40T + 1.33e3T^{2} \) |
| 13 | \( 1 + 12iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 58iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 26T + 6.85e3T^{2} \) |
| 23 | \( 1 + 64iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 62T + 2.43e4T^{2} \) |
| 31 | \( 1 - 252T + 2.97e4T^{2} \) |
| 37 | \( 1 + 26iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 6T + 6.89e4T^{2} \) |
| 43 | \( 1 - 416iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 396iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 450iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 274T + 2.05e5T^{2} \) |
| 61 | \( 1 + 576T + 2.26e5T^{2} \) |
| 67 | \( 1 - 476iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 448T + 3.57e5T^{2} \) |
| 73 | \( 1 + 158iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 936T + 4.93e5T^{2} \) |
| 83 | \( 1 - 530iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 390T + 7.04e5T^{2} \) |
| 97 | \( 1 + 214iT - 9.12e5T^{2} \) |
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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
−10.26354077650510876874981266995, −8.873280560540529489755247629587, −8.079727495995427160459049558009, −7.69168387454015940292331057191, −6.46989997938655253818042128523, −6.04576614918098085490146016985, −4.88898602165617964652915421552, −2.99920736573025120404062241493, −2.24857804006270307296183536801, −1.05294842235587658032454437375,
0.21456077726332072415174863030, 2.59870392667337996769121661753, 3.51519055349033545532931939871, 4.56765587574096016313135081369, 5.11905280327120957863789628017, 6.13939448155134037862573833905, 7.54676023034177350406559319213, 8.542009338534408257723824179384, 9.283676535122850171589696331540, 10.17303113598695881277814355869
