| L(s) = 1 | + (−1.5 + 0.866i)3-s + (−2 − 1.73i)7-s + (1.5 − 2.59i)9-s + 5.19i·11-s + 1.73i·13-s + 3·17-s − 3.46i·19-s + (4.5 + 0.866i)21-s + 5.19i·27-s + 5.19i·29-s − 10.3i·31-s + (−4.5 − 7.79i)33-s + 8·37-s + (−1.49 − 2.59i)39-s − 6·41-s + ⋯ |
| L(s) = 1 | + (−0.866 + 0.499i)3-s + (−0.755 − 0.654i)7-s + (0.5 − 0.866i)9-s + 1.56i·11-s + 0.480i·13-s + 0.727·17-s − 0.794i·19-s + (0.981 + 0.188i)21-s + 0.999i·27-s + 0.964i·29-s − 1.86i·31-s + (−0.783 − 1.35i)33-s + 1.31·37-s + (−0.240 − 0.416i)39-s − 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 - 0.188i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.981 - 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3535295654\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3535295654\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.5 - 0.866i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2 + 1.73i)T \) |
| good | 11 | \( 1 - 5.19iT - 11T^{2} \) |
| 13 | \( 1 - 1.73iT - 13T^{2} \) |
| 17 | \( 1 - 3T + 17T^{2} \) |
| 19 | \( 1 + 3.46iT - 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 5.19iT - 29T^{2} \) |
| 31 | \( 1 + 10.3iT - 31T^{2} \) |
| 37 | \( 1 - 8T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 10T + 43T^{2} \) |
| 47 | \( 1 - 3T + 47T^{2} \) |
| 53 | \( 1 - 10.3iT - 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 - 6.92iT - 61T^{2} \) |
| 67 | \( 1 + 2T + 67T^{2} \) |
| 71 | \( 1 - 10.3iT - 71T^{2} \) |
| 73 | \( 1 + 6.92iT - 73T^{2} \) |
| 79 | \( 1 + 13T + 79T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 1.73iT - 97T^{2} \) |
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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
−9.729184435971194510321989448901, −9.008262180957748152244164919088, −7.61912106993903455734081943332, −7.04672926578760164840412656180, −6.39270562968678152146015670687, −5.46660171347011354232674929558, −4.50769752810208865971432663105, −4.04536584362024925044513549539, −2.83211408681181533885687999866, −1.33488803561222162646106126109,
0.15278607508581560021782569393, 1.41982939658653832192944891929, 2.85584658164941349263425279384, 3.60783018052479318317420373352, 5.02935636987606817124427535089, 5.73756902911413393800885070685, 6.20581759718519006484921154544, 6.99258156101152089273126480666, 8.127657355138364789031369488175, 8.463213498431155976363988716114
