| L(s) = 1 | + i·3-s + 4i·7-s − 9-s + 6i·13-s + 2i·17-s − 4·19-s − 4·21-s + 8i·23-s − i·27-s − 6·29-s − 6i·37-s − 6·39-s + 10·41-s − 4i·43-s + 8i·47-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 1.51i·7-s − 0.333·9-s + 1.66i·13-s + 0.485i·17-s − 0.917·19-s − 0.872·21-s + 1.66i·23-s − 0.192i·27-s − 1.11·29-s − 0.986i·37-s − 0.960·39-s + 1.56·41-s − 0.609i·43-s + 1.16i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.113481484\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.113481484\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 4iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 6iT - 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 - 8iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 + 10iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 14iT - 73T^{2} \) |
| 79 | \( 1 - 16T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−9.027889241861959885587486173493, −8.095145369001344704414318416768, −7.28500933842082239871001488842, −6.28633067041956117914708403611, −5.81690269783928820787763513248, −5.06493257573387129424754687241, −4.18252272051402475855765849818, −3.52216754910067126398048397586, −2.33005479422131672625401914617, −1.79099712360027512581802621209,
0.32704372559597997359209553918, 1.05011336400021207754614972501, 2.36712647095184738611668133738, 3.20983558407158514714468589104, 4.12770966459033456279726101144, 4.83230405348179457841508999607, 5.82105543763721700221125087307, 6.51288879626561545254674699794, 7.23130589566269958221591522735, 7.81612587206599255781830128993
