| 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 + (−1.50 − 4.33i)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 + (−0.327 − 0.944i)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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.327 + 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.327 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.752162663\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.752162663\) |
| \(L(\frac{3}{2})\) |
|
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
−9.036412174497520748772441397748, −8.254262014605625558225180004215, −7.56369709250770423541135314496, −6.58718638573657582280706707464, −5.90620403807445562978891439496, −4.63041175092494964140998678638, −3.94049459190976252388460318175, −3.14585084964157272850217713326, −2.26258112374088369481753188021, −0.53952887172912096435875958924,
1.48714588520535865857193686221, 2.45582090668844393392569094598, 3.24823513247472759041337090653, 4.26363001934459925378921122715, 5.26428175081447042361709621032, 6.38860936045813883127184559940, 6.93559735776402338349003219286, 7.70144871299683551441314461885, 8.549008654602721632413392486782, 9.181494373520389908178813128125
