| 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 | \( 1 + 1.73iT - 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.181494373520389908178813128125, −8.549008654602721632413392486782, −7.70144871299683551441314461885, −6.93559735776402338349003219286, −6.38860936045813883127184559940, −5.26428175081447042361709621032, −4.26363001934459925378921122715, −3.24823513247472759041337090653, −2.45582090668844393392569094598, −1.48714588520535865857193686221,
0.53952887172912096435875958924, 2.26258112374088369481753188021, 3.14585084964157272850217713326, 3.94049459190976252388460318175, 4.63041175092494964140998678638, 5.90620403807445562978891439496, 6.58718638573657582280706707464, 7.56369709250770423541135314496, 8.254262014605625558225180004215, 9.036412174497520748772441397748
