L(s) = 1 | + (0.618 + 1.27i)2-s + (2.16 − 2.16i)3-s + (−1.23 + 1.57i)4-s + (4.09 + 1.41i)6-s − 3.30i·7-s + (−2.76 − 0.594i)8-s − 6.40i·9-s + (2.01 + 2.01i)11-s + (0.738 + 6.08i)12-s + (0.794 − 0.794i)13-s + (4.20 − 2.04i)14-s + (−0.955 − 3.88i)16-s + 4.61·17-s + (8.14 − 3.96i)18-s + (−3.48 + 3.48i)19-s + ⋯ |
L(s) = 1 | + (0.437 + 0.899i)2-s + (1.25 − 1.25i)3-s + (−0.616 + 0.787i)4-s + (1.67 + 0.577i)6-s − 1.24i·7-s + (−0.977 − 0.210i)8-s − 2.13i·9-s + (0.606 + 0.606i)11-s + (0.213 + 1.75i)12-s + (0.220 − 0.220i)13-s + (1.12 − 0.546i)14-s + (−0.238 − 0.971i)16-s + 1.11·17-s + (1.91 − 0.934i)18-s + (−0.800 + 0.800i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.150i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 + 0.150i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.32553 - 0.176426i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.32553 - 0.176426i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.618 - 1.27i)T \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (-2.16 + 2.16i)T - 3iT^{2} \) |
| 7 | \( 1 + 3.30iT - 7T^{2} \) |
| 11 | \( 1 + (-2.01 - 2.01i)T + 11iT^{2} \) |
| 13 | \( 1 + (-0.794 + 0.794i)T - 13iT^{2} \) |
| 17 | \( 1 - 4.61T + 17T^{2} \) |
| 19 | \( 1 + (3.48 - 3.48i)T - 19iT^{2} \) |
| 23 | \( 1 - 7.99iT - 23T^{2} \) |
| 29 | \( 1 + (1.95 - 1.95i)T - 29iT^{2} \) |
| 31 | \( 1 + 5.12T + 31T^{2} \) |
| 37 | \( 1 + (-0.448 - 0.448i)T + 37iT^{2} \) |
| 41 | \( 1 + 4.02iT - 41T^{2} \) |
| 43 | \( 1 + (-4.97 - 4.97i)T + 43iT^{2} \) |
| 47 | \( 1 + 5.49T + 47T^{2} \) |
| 53 | \( 1 + (3.35 + 3.35i)T + 53iT^{2} \) |
| 59 | \( 1 + (-2.07 - 2.07i)T + 59iT^{2} \) |
| 61 | \( 1 + (0.557 - 0.557i)T - 61iT^{2} \) |
| 67 | \( 1 + (0.636 - 0.636i)T - 67iT^{2} \) |
| 71 | \( 1 - 6.85iT - 71T^{2} \) |
| 73 | \( 1 - 10.5iT - 73T^{2} \) |
| 79 | \( 1 + 17.3T + 79T^{2} \) |
| 83 | \( 1 + (-9.48 + 9.48i)T - 83iT^{2} \) |
| 89 | \( 1 + 7.62iT - 89T^{2} \) |
| 97 | \( 1 - 0.709T + 97T^{2} \) |
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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
−11.63849355716918562347042659683, −9.985582341825470281068818910501, −9.094469857015280754989866431899, −8.055449650290741883392958466455, −7.46130775559178966801392262621, −6.92505911191402201517902187369, −5.78055972464886954248251957107, −4.02918159587653277555652378028, −3.33387143943533306356645488013, −1.47824700453599774792846152482,
2.22740689906662645881754816442, 3.11469625764528404665319218582, 4.08032327217508791211754541046, 5.06547544521573756326770544040, 6.16313533229818479075551098089, 8.219135892870691865115758169596, 8.952344730977179624703490114384, 9.349689209784327360043172005609, 10.39210601331403996358160532809, 11.13049923380822064836619622570