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
−11.13049923380822064836619622570, −10.39210601331403996358160532809, −9.349689209784327360043172005609, −8.952344730977179624703490114384, −8.219135892870691865115758169596, −6.16313533229818479075551098089, −5.06547544521573756326770544040, −4.08032327217508791211754541046, −3.11469625764528404665319218582, −2.22740689906662645881754816442,
1.47824700453599774792846152482, 3.33387143943533306356645488013, 4.02918159587653277555652378028, 5.78055972464886954248251957107, 6.92505911191402201517902187369, 7.46130775559178966801392262621, 8.055449650290741883392958466455, 9.094469857015280754989866431899, 9.985582341825470281068818910501, 11.63849355716918562347042659683