| L(s) = 1 | + 3.82·5-s − 7-s − 0.828·11-s − 2.82·13-s − 5.82·17-s − 1.17·19-s − 4·23-s + 9.65·25-s + 2.82·29-s + 1.17·31-s − 3.82·35-s − 8.65·37-s + 1.82·41-s − 2.17·43-s − 2.65·47-s + 49-s + 2·53-s − 3.17·55-s − 10.6·59-s + 8.82·61-s − 10.8·65-s − 4·67-s − 9.65·71-s + 5.65·73-s + 0.828·77-s − 3.82·79-s − 16.6·83-s + ⋯ |
| L(s) = 1 | + 1.71·5-s − 0.377·7-s − 0.249·11-s − 0.784·13-s − 1.41·17-s − 0.268·19-s − 0.834·23-s + 1.93·25-s + 0.525·29-s + 0.210·31-s − 0.647·35-s − 1.42·37-s + 0.285·41-s − 0.331·43-s − 0.387·47-s + 0.142·49-s + 0.274·53-s − 0.427·55-s − 1.38·59-s + 1.13·61-s − 1.34·65-s − 0.488·67-s − 1.14·71-s + 0.662·73-s + 0.0944·77-s − 0.430·79-s − 1.82·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| good | 5 | \( 1 - 3.82T + 5T^{2} \) |
| 11 | \( 1 + 0.828T + 11T^{2} \) |
| 13 | \( 1 + 2.82T + 13T^{2} \) |
| 17 | \( 1 + 5.82T + 17T^{2} \) |
| 19 | \( 1 + 1.17T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 2.82T + 29T^{2} \) |
| 31 | \( 1 - 1.17T + 31T^{2} \) |
| 37 | \( 1 + 8.65T + 37T^{2} \) |
| 41 | \( 1 - 1.82T + 41T^{2} \) |
| 43 | \( 1 + 2.17T + 43T^{2} \) |
| 47 | \( 1 + 2.65T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 + 10.6T + 59T^{2} \) |
| 61 | \( 1 - 8.82T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 9.65T + 71T^{2} \) |
| 73 | \( 1 - 5.65T + 73T^{2} \) |
| 79 | \( 1 + 3.82T + 79T^{2} \) |
| 83 | \( 1 + 16.6T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 10.1T + 97T^{2} \) |
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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
−7.65396321981258921492626199941, −6.72119574018084756204827585770, −6.41225230463595037920270164823, −5.59367111080478406704448823629, −4.97586812430340030043080397034, −4.17049172428244662100081706262, −2.91984845966081298353489610819, −2.29742230343259759493149632541, −1.58298195948073124474948445394, 0,
1.58298195948073124474948445394, 2.29742230343259759493149632541, 2.91984845966081298353489610819, 4.17049172428244662100081706262, 4.97586812430340030043080397034, 5.59367111080478406704448823629, 6.41225230463595037920270164823, 6.72119574018084756204827585770, 7.65396321981258921492626199941
