L(s) = 1 | − 1.75i·5-s − 7-s + 1.75i·11-s + 1.46i·13-s − 6.54·17-s − 3.46i·19-s + 3.03·23-s + 1.92·25-s + 9.57i·29-s + 2·31-s + 1.75i·35-s + 4.53i·37-s + 6.54·41-s − 4.92i·43-s + 9.57·47-s + ⋯ |
L(s) = 1 | − 0.783i·5-s − 0.377·7-s + 0.528i·11-s + 0.406i·13-s − 1.58·17-s − 0.794i·19-s + 0.632·23-s + 0.385·25-s + 1.77i·29-s + 0.359·31-s + 0.296i·35-s + 0.745i·37-s + 1.02·41-s − 0.751i·43-s + 1.39·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.603997588\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.603997588\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.75iT - 5T^{2} \) |
| 11 | \( 1 - 1.75iT - 11T^{2} \) |
| 13 | \( 1 - 1.46iT - 13T^{2} \) |
| 17 | \( 1 + 6.54T + 17T^{2} \) |
| 19 | \( 1 + 3.46iT - 19T^{2} \) |
| 23 | \( 1 - 3.03T + 23T^{2} \) |
| 29 | \( 1 - 9.57iT - 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 - 4.53iT - 37T^{2} \) |
| 41 | \( 1 - 6.54T + 41T^{2} \) |
| 43 | \( 1 + 4.92iT - 43T^{2} \) |
| 47 | \( 1 - 9.57T + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 9.57iT - 59T^{2} \) |
| 61 | \( 1 + 1.46iT - 61T^{2} \) |
| 67 | \( 1 + 10iT - 67T^{2} \) |
| 71 | \( 1 + 10.0T + 71T^{2} \) |
| 73 | \( 1 - 12.9T + 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 - 6.07iT - 83T^{2} \) |
| 89 | \( 1 - 0.469T + 89T^{2} \) |
| 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
−8.742910714629376942988699186430, −7.62836092466549002736922340574, −6.82588023576289299170692321519, −6.43478548871952444788102430996, −5.14091164778856180074884602748, −4.79059088651297562900305536565, −3.93820964405212097450547681141, −2.84433144646388543610649909206, −1.91670619290867144160380309885, −0.71701773634126550084430128492,
0.71173393842732793054112176960, 2.28050960033743704528443651185, 2.87697583361904601569520022758, 3.86518059780087816852621009487, 4.56370317308252472730133031475, 5.79571200472151458310260578900, 6.18761294675074663992976223911, 7.01468687402680562088245453709, 7.64628165976651142812148658267, 8.513385710136700887504165647525