| L(s) = 1 | + 3.46i·7-s − 3.46i·11-s + 3.46i·13-s + 6·17-s + (−4 − 1.73i)19-s − 5·25-s + 6.92i·29-s + 10·31-s + 3.46i·37-s − 6.92i·41-s + 10.3i·43-s − 6.92i·47-s − 4.99·49-s + 13.8i·53-s − 12·59-s + ⋯ |
| L(s) = 1 | + 1.30i·7-s − 1.04i·11-s + 0.960i·13-s + 1.45·17-s + (−0.917 − 0.397i)19-s − 25-s + 1.28i·29-s + 1.79·31-s + 0.569i·37-s − 1.08i·41-s + 1.58i·43-s − 1.01i·47-s − 0.714·49-s + 1.90i·53-s − 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.114 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.114 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.484631991\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.484631991\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (4 + 1.73i)T \) |
| good | 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 - 3.46iT - 7T^{2} \) |
| 11 | \( 1 + 3.46iT - 11T^{2} \) |
| 13 | \( 1 - 3.46iT - 13T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 6.92iT - 29T^{2} \) |
| 31 | \( 1 - 10T + 31T^{2} \) |
| 37 | \( 1 - 3.46iT - 37T^{2} \) |
| 41 | \( 1 + 6.92iT - 41T^{2} \) |
| 43 | \( 1 - 10.3iT - 43T^{2} \) |
| 47 | \( 1 + 6.92iT - 47T^{2} \) |
| 53 | \( 1 - 13.8iT - 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 83 | \( 1 - 3.46iT - 83T^{2} \) |
| 89 | \( 1 - 6.92iT - 89T^{2} \) |
| 97 | \( 1 - 6.92iT - 97T^{2} \) |
| 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
−8.880406196379438251065378819208, −8.457415184628673389608811903699, −7.63843402015391526501072709137, −6.53955395913790917394545835636, −5.97645096262885397519981471185, −5.28238151014521328625690719694, −4.31941517583379993633591418436, −3.24018838777481419141260908090, −2.48747670729328495624732126227, −1.29353376480147041091862986209,
0.50757630156230118219999669991, 1.70054142344014590285747424452, 2.93870493198868333070785605224, 3.95720528457469954360782911870, 4.50119585284574641246200022451, 5.56587821216796220017967540132, 6.34896869458396410845913559818, 7.27029960141297032016128915688, 7.83097810479085325127171781506, 8.341840739673739455667916624142
