| L(s) = 1 | + i·2-s − 4-s + (1.41 + 2.23i)7-s − i·8-s − 5.65i·11-s + 4.47i·13-s + (−2.23 + 1.41i)14-s + 16-s − 3.16·17-s − 3.16i·19-s + 5.65·22-s + 4i·23-s − 4.47·26-s + (−1.41 − 2.23i)28-s + 2.82i·29-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s + (0.534 + 0.845i)7-s − 0.353i·8-s − 1.70i·11-s + 1.24i·13-s + (−0.597 + 0.377i)14-s + 0.250·16-s − 0.766·17-s − 0.725i·19-s + 1.20·22-s + 0.834i·23-s − 0.877·26-s + (−0.267 − 0.422i)28-s + 0.525i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.924 - 0.381i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.924 - 0.381i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.180540751\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.180540751\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-1.41 - 2.23i)T \) |
| good | 11 | \( 1 + 5.65iT - 11T^{2} \) |
| 13 | \( 1 - 4.47iT - 13T^{2} \) |
| 17 | \( 1 + 3.16T + 17T^{2} \) |
| 19 | \( 1 + 3.16iT - 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 - 2.82iT - 29T^{2} \) |
| 31 | \( 1 - 6.32iT - 31T^{2} \) |
| 37 | \( 1 + 9.89T + 37T^{2} \) |
| 41 | \( 1 - 4.47T + 41T^{2} \) |
| 43 | \( 1 - 1.41T + 43T^{2} \) |
| 47 | \( 1 - 9.48T + 47T^{2} \) |
| 53 | \( 1 - 4iT - 53T^{2} \) |
| 59 | \( 1 + 4.47T + 59T^{2} \) |
| 61 | \( 1 - 9.48iT - 61T^{2} \) |
| 67 | \( 1 + 7.07T + 67T^{2} \) |
| 71 | \( 1 + 1.41iT - 71T^{2} \) |
| 73 | \( 1 - 13.4iT - 73T^{2} \) |
| 79 | \( 1 + 6T + 79T^{2} \) |
| 83 | \( 1 - 12.6T + 83T^{2} \) |
| 89 | \( 1 - 4.47T + 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−8.949147708769286615102760815653, −8.462429120465982497980830022377, −7.43624913594964837591643980088, −6.74704264881584829727653251325, −5.95746335514814381120497885003, −5.35246834374156482319259120934, −4.54402847928347666182217462750, −3.57420189557019447660713647645, −2.55541626914829658500895705423, −1.30234011545016373952410581256,
0.37633734486404252227810484830, 1.69307520812416241718027058058, 2.45333222580209387421501968553, 3.67235744003756070148238357421, 4.40071967815586469699597783551, 4.97572789176488853186579991074, 6.00391566480026714858565449624, 7.05521085061332902229345656169, 7.68508122227383913003611988246, 8.293283353731506477867108411592
