| L(s) = 1 | + 0.399·5-s − 9.64·7-s − 9.57·11-s + 9.40i·13-s + 25.3·17-s + (1.88 + 18.9i)19-s − 14.2·23-s − 24.8·25-s − 10.9i·29-s − 12.8i·31-s − 3.85·35-s + 3.29i·37-s + 63.6i·41-s + 26.8·43-s − 18.1·47-s + ⋯ |
| L(s) = 1 | + 0.0799·5-s − 1.37·7-s − 0.870·11-s + 0.723i·13-s + 1.49·17-s + (0.0989 + 0.995i)19-s − 0.621·23-s − 0.993·25-s − 0.377i·29-s − 0.415i·31-s − 0.110·35-s + 0.0890i·37-s + 1.55i·41-s + 0.625·43-s − 0.386·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0989 + 0.995i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0989 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7367854176\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7367854176\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (-1.88 - 18.9i)T \) |
| good | 5 | \( 1 - 0.399T + 25T^{2} \) |
| 7 | \( 1 + 9.64T + 49T^{2} \) |
| 11 | \( 1 + 9.57T + 121T^{2} \) |
| 13 | \( 1 - 9.40iT - 169T^{2} \) |
| 17 | \( 1 - 25.3T + 289T^{2} \) |
| 23 | \( 1 + 14.2T + 529T^{2} \) |
| 29 | \( 1 + 10.9iT - 841T^{2} \) |
| 31 | \( 1 + 12.8iT - 961T^{2} \) |
| 37 | \( 1 - 3.29iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 63.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 26.8T + 1.84e3T^{2} \) |
| 47 | \( 1 + 18.1T + 2.20e3T^{2} \) |
| 53 | \( 1 + 36.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 12.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 46.8T + 3.72e3T^{2} \) |
| 67 | \( 1 - 71.4iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 36.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 60.6T + 5.32e3T^{2} \) |
| 79 | \( 1 + 101. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 63.1T + 6.88e3T^{2} \) |
| 89 | \( 1 + 107. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 120. iT - 9.40e3T^{2} \) |
| 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
−8.343037786268574217285593248572, −7.75543407315443381175898454673, −6.96093587835842569354299858315, −5.95068358599723384057311433323, −5.72369744870759072972198266978, −4.41438547043571780942120922754, −3.55036497673705969769709637680, −2.83012559788207017498579012812, −1.67201791354769404312312019091, −0.22305541379725383830432910080,
0.792122129378495490042380955822, 2.36341323711540185551520132090, 3.16220654681430952069650422547, 3.83703523271598637459160984916, 5.16757306722299989556862983780, 5.69907685572739932829483070671, 6.49336341271165084021670585989, 7.41109409580523866785779496790, 7.917017765807441357712778664134, 8.934251865085004885000315709688
