| L(s) = 1 | + 3.15i·3-s + 2.09i·5-s + (−1.59 − 2.10i)7-s − 6.92·9-s + (1.48 − 2.96i)11-s − 13-s − 6.61·15-s − 5.21·17-s + 3.50·19-s + (6.64 − 5.03i)21-s − 3.97·23-s + 0.593·25-s − 12.3i·27-s − 0.171i·29-s − 6.22i·31-s + ⋯ |
| L(s) = 1 | + 1.81i·3-s + 0.938i·5-s + (−0.604 − 0.796i)7-s − 2.30·9-s + (0.447 − 0.894i)11-s − 0.277·13-s − 1.70·15-s − 1.26·17-s + 0.804·19-s + (1.44 − 1.09i)21-s − 0.828·23-s + 0.118·25-s − 2.37i·27-s − 0.0317i·29-s − 1.11i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.184i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.982 - 0.184i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9463161118\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9463161118\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (1.59 + 2.10i)T \) |
| 11 | \( 1 + (-1.48 + 2.96i)T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 3.15iT - 3T^{2} \) |
| 5 | \( 1 - 2.09iT - 5T^{2} \) |
| 17 | \( 1 + 5.21T + 17T^{2} \) |
| 19 | \( 1 - 3.50T + 19T^{2} \) |
| 23 | \( 1 + 3.97T + 23T^{2} \) |
| 29 | \( 1 + 0.171iT - 29T^{2} \) |
| 31 | \( 1 + 6.22iT - 31T^{2} \) |
| 37 | \( 1 + 9.42T + 37T^{2} \) |
| 41 | \( 1 - 5.25T + 41T^{2} \) |
| 43 | \( 1 + 2.92iT - 43T^{2} \) |
| 47 | \( 1 - 9.40iT - 47T^{2} \) |
| 53 | \( 1 - 5.09T + 53T^{2} \) |
| 59 | \( 1 + 9.32iT - 59T^{2} \) |
| 61 | \( 1 - 0.809T + 61T^{2} \) |
| 67 | \( 1 - 4.09T + 67T^{2} \) |
| 71 | \( 1 - 10.2T + 71T^{2} \) |
| 73 | \( 1 - 15.7T + 73T^{2} \) |
| 79 | \( 1 - 2.13iT - 79T^{2} \) |
| 83 | \( 1 - 4.96T + 83T^{2} \) |
| 89 | \( 1 + 15.0iT - 89T^{2} \) |
| 97 | \( 1 - 1.01iT - 97T^{2} \) |
| 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
−8.709745395653753098409714414877, −7.82456778114422226678612512316, −6.82966204548324666644445895724, −6.23117659560609822334441361646, −5.41559091458776516955276897362, −4.47901565877073018845484695114, −3.78522805384617844880782085785, −3.32955623942370823853770351707, −2.44740362725861338955121105265, −0.32173059619967853945626759224,
0.924165850227593843935709484322, 1.93290250118827783129254737037, 2.47299506128791319085259594729, 3.68801548173209979966980420634, 4.98067369616654091254562437350, 5.49992326889122536439655522647, 6.57560131949466833303874606004, 6.78202447860411990750257267891, 7.63703393245647544568715773129, 8.442752809799729760105758777537
