| L(s) = 1 | + 2.82·5-s + (2.44 + i)7-s + 3.46·11-s + 3.00·25-s + 10.3i·29-s + 4.89·31-s + (6.92 + 2.82i)35-s + (4.99 + 4.89i)49-s − 3.46i·53-s + 9.79·55-s − 11.3i·59-s + 9.79i·73-s + (8.48 + 3.46i)77-s − 10i·79-s − 5.65i·83-s + ⋯ |
| L(s) = 1 | + 1.26·5-s + (0.925 + 0.377i)7-s + 1.04·11-s + 0.600·25-s + 1.92i·29-s + 0.879·31-s + (1.17 + 0.478i)35-s + (0.714 + 0.699i)49-s − 0.475i·53-s + 1.32·55-s − 1.47i·59-s + 1.14i·73-s + (0.966 + 0.394i)77-s − 1.12i·79-s − 0.620i·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.921 - 0.387i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.921 - 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.090006877\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.090006877\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.44 - i)T \) |
| good | 5 | \( 1 - 2.82T + 5T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 10.3iT - 29T^{2} \) |
| 31 | \( 1 - 4.89T + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 3.46iT - 53T^{2} \) |
| 59 | \( 1 + 11.3iT - 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 - 9.79iT - 73T^{2} \) |
| 79 | \( 1 + 10iT - 79T^{2} \) |
| 83 | \( 1 + 5.65iT - 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 19.5iT - 97T^{2} \) |
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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
−8.651118435546518761872885873202, −7.83801105569528023830436876058, −6.82403109860327361110252143200, −6.32552175909732720044882215198, −5.43257324244942868918999172969, −4.95095124586192884167671849206, −3.94681245089102880174134105378, −2.85351798921936356053279622551, −1.87137014662702087244825124698, −1.25676149279013196196994183596,
1.00794627169091646984386145154, 1.84100185599635969594692002599, 2.67232138502362879615783706241, 3.98683167670790432418336543091, 4.57062957692753102987189685376, 5.54948116788909393405525358844, 6.12921880322043099919679818491, 6.82664001011430475779761028812, 7.70784116635654382832031236199, 8.424370292505083096620465437149
