L(s) = 1 | − i·3-s − 3.46i·5-s − 7-s − 9-s − 1.46i·11-s + 2i·13-s − 3.46·15-s + 0.535·17-s − 6.92i·19-s + i·21-s − 1.46·23-s − 6.99·25-s + i·27-s − 4.92i·29-s + 10.9·31-s + ⋯ |
L(s) = 1 | − 0.577i·3-s − 1.54i·5-s − 0.377·7-s − 0.333·9-s − 0.441i·11-s + 0.554i·13-s − 0.894·15-s + 0.129·17-s − 1.58i·19-s + 0.218i·21-s − 0.305·23-s − 1.39·25-s + 0.192i·27-s − 0.915i·29-s + 1.96·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9067099658\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9067099658\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 7 | \( 1 + T \) |
good | 5 | \( 1 + 3.46iT - 5T^{2} \) |
| 11 | \( 1 + 1.46iT - 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 - 0.535T + 17T^{2} \) |
| 19 | \( 1 + 6.92iT - 19T^{2} \) |
| 23 | \( 1 + 1.46T + 23T^{2} \) |
| 29 | \( 1 + 4.92iT - 29T^{2} \) |
| 31 | \( 1 - 10.9T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 11.4T + 41T^{2} \) |
| 43 | \( 1 + 8iT - 43T^{2} \) |
| 47 | \( 1 + 10.9T + 47T^{2} \) |
| 53 | \( 1 - 2iT - 53T^{2} \) |
| 59 | \( 1 + 1.07iT - 59T^{2} \) |
| 61 | \( 1 + 8.92iT - 61T^{2} \) |
| 67 | \( 1 - 2.92iT - 67T^{2} \) |
| 71 | \( 1 - 9.46T + 71T^{2} \) |
| 73 | \( 1 + 12.9T + 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 + 3.46T + 89T^{2} \) |
| 97 | \( 1 + 8.92T + 97T^{2} \) |
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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
−8.099566506330015599690401524290, −6.79712304551969916827454066840, −6.49473822275581776539006416941, −5.44368171977055426654156620531, −4.87856517291846555325215162217, −4.16560479525379401625401012912, −3.09028291474954499033122329197, −2.09873858791660656118434884941, −1.08050424058809366856112584876, −0.25556661915958357521208178094,
1.63695664133786720557699602520, 2.83159168981987187148819624927, 3.24873617873961293269249022173, 4.03077924771426807265346827817, 4.99119089105707451068748336373, 5.89857342104549549711355509757, 6.47040478312352567684015844941, 7.08526760634340866410303045111, 7.971384192490741259227325882667, 8.428896538326947298821535800810