| L(s) = 1 | + i·3-s + (−1.75 − 1.38i)5-s − 9-s + 1.50·11-s + i·13-s + (1.38 − 1.75i)15-s + 2.72i·17-s − 0.726·19-s − 4.72i·23-s + (1.14 + 4.86i)25-s − i·27-s + 7.55·29-s − 3.00·31-s + 1.50i·33-s + 5.00i·37-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + (−0.783 − 0.621i)5-s − 0.333·9-s + 0.453·11-s + 0.277i·13-s + (0.358 − 0.452i)15-s + 0.661i·17-s − 0.166·19-s − 0.985i·23-s + (0.228 + 0.973i)25-s − 0.192i·27-s + 1.40·29-s − 0.540·31-s + 0.261i·33-s + 0.823i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.621 - 0.783i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.621 - 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.341494545\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.341494545\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.75 + 1.38i)T \) |
| 13 | \( 1 - iT \) |
| good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 1.50T + 11T^{2} \) |
| 17 | \( 1 - 2.72iT - 17T^{2} \) |
| 19 | \( 1 + 0.726T + 19T^{2} \) |
| 23 | \( 1 + 4.72iT - 23T^{2} \) |
| 29 | \( 1 - 7.55T + 29T^{2} \) |
| 31 | \( 1 + 3.00T + 31T^{2} \) |
| 37 | \( 1 - 5.00iT - 37T^{2} \) |
| 41 | \( 1 - 5.78T + 41T^{2} \) |
| 43 | \( 1 - 2.72iT - 43T^{2} \) |
| 47 | \( 1 - 10.2iT - 47T^{2} \) |
| 53 | \( 1 + 7.55iT - 53T^{2} \) |
| 59 | \( 1 - 12.5T + 59T^{2} \) |
| 61 | \( 1 - 6.28T + 61T^{2} \) |
| 67 | \( 1 - 12.5iT - 67T^{2} \) |
| 71 | \( 1 - 4.77T + 71T^{2} \) |
| 73 | \( 1 - 12.0iT - 73T^{2} \) |
| 79 | \( 1 + 5.27T + 79T^{2} \) |
| 83 | \( 1 - 7.78iT - 83T^{2} \) |
| 89 | \( 1 + 1.78T + 89T^{2} \) |
| 97 | \( 1 - 6iT - 97T^{2} \) |
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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
−9.529483734734021801915179388985, −8.555382327827838305792906697918, −8.330674430023863386139069887098, −7.15077712551556995085925016431, −6.30550584023024407782859623276, −5.26696690886148509486899115733, −4.36223527195295732273681589980, −3.87403364941065329325446591857, −2.62788764105150553599212179629, −1.02576401867867564325747385002,
0.67100685482534905698289186302, 2.22509669275629148811609855287, 3.25242509453202029788493913655, 4.10730708062395728568942631491, 5.26525440344810809138777868042, 6.23384760064478230719889467819, 7.08525839282254982372441470937, 7.54117328949581074182351928452, 8.436693432652361115141152991773, 9.192065201062430856056753813061
