L(s) = 1 | − 1.75i·5-s + 7-s − 1.75i·11-s − 1.46i·13-s + 6.54·17-s − 3.46i·19-s + 3.03·23-s + 1.92·25-s + 9.57i·29-s − 2·31-s − 1.75i·35-s − 4.53i·37-s − 6.54·41-s − 4.92i·43-s + 9.57·47-s + ⋯ |
L(s) = 1 | − 0.783i·5-s + 0.377·7-s − 0.528i·11-s − 0.406i·13-s + 1.58·17-s − 0.794i·19-s + 0.632·23-s + 0.385·25-s + 1.77i·29-s − 0.359·31-s − 0.296i·35-s − 0.745i·37-s − 1.02·41-s − 0.751i·43-s + 1.39·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.071492528\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.071492528\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 5 | \( 1 + 1.75iT - 5T^{2} \) |
| 11 | \( 1 + 1.75iT - 11T^{2} \) |
| 13 | \( 1 + 1.46iT - 13T^{2} \) |
| 17 | \( 1 - 6.54T + 17T^{2} \) |
| 19 | \( 1 + 3.46iT - 19T^{2} \) |
| 23 | \( 1 - 3.03T + 23T^{2} \) |
| 29 | \( 1 - 9.57iT - 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 + 4.53iT - 37T^{2} \) |
| 41 | \( 1 + 6.54T + 41T^{2} \) |
| 43 | \( 1 + 4.92iT - 43T^{2} \) |
| 47 | \( 1 - 9.57T + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 - 9.57iT - 59T^{2} \) |
| 61 | \( 1 - 1.46iT - 61T^{2} \) |
| 67 | \( 1 + 10iT - 67T^{2} \) |
| 71 | \( 1 + 10.0T + 71T^{2} \) |
| 73 | \( 1 - 12.9T + 73T^{2} \) |
| 79 | \( 1 + 10.9T + 79T^{2} \) |
| 83 | \( 1 + 6.07iT - 83T^{2} \) |
| 89 | \( 1 + 0.469T + 89T^{2} \) |
| 97 | \( 1 - 8.92T + 97T^{2} \) |
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
−8.535842182111023980654386537953, −7.49264810622283466684977964407, −7.04608338140374027118921134719, −5.80834395889938822948697217694, −5.31438764237845130390018347631, −4.68999886626925492586048994302, −3.58359195321088109232147032707, −2.88589963536238719480019553893, −1.52109154063351859705576581957, −0.69283111625337526211769927493,
1.15793531646292830672083209286, 2.22491404190014829906782816706, 3.14816172743169820047051959577, 3.94458672611087557053484062582, 4.85981983769179789979507403195, 5.67781140685535646483900053912, 6.42807092926037638846607137847, 7.20488709376725614423160682505, 7.79036632329641421266535758418, 8.443327078504079642961903941604