L(s) = 1 | + (−0.856 − 1.12i)2-s − 3-s + (−0.533 + 1.92i)4-s + 3.85i·5-s + (0.856 + 1.12i)6-s + (2.62 − 1.05i)8-s + 9-s + (4.33 − 3.30i)10-s − 1.36i·11-s + (0.533 − 1.92i)12-s − 0.369i·13-s − 3.85i·15-s + (−3.43 − 2.05i)16-s + 4.50i·17-s + (−0.856 − 1.12i)18-s + 0.0661·19-s + ⋯ |
L(s) = 1 | + (−0.605 − 0.795i)2-s − 0.577·3-s + (−0.266 + 0.963i)4-s + 1.72i·5-s + (0.349 + 0.459i)6-s + (0.928 − 0.371i)8-s + 0.333·9-s + (1.37 − 1.04i)10-s − 0.410i·11-s + (0.153 − 0.556i)12-s − 0.102i·13-s − 0.995i·15-s + (−0.857 − 0.513i)16-s + 1.09i·17-s + (−0.201 − 0.265i)18-s + 0.0151·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.429 - 0.903i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.429 - 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.265874 + 0.420852i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.265874 + 0.420852i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.856 + 1.12i)T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 3.85iT - 5T^{2} \) |
| 11 | \( 1 + 1.36iT - 11T^{2} \) |
| 13 | \( 1 + 0.369iT - 13T^{2} \) |
| 17 | \( 1 - 4.50iT - 17T^{2} \) |
| 19 | \( 1 - 0.0661T + 19T^{2} \) |
| 23 | \( 1 - 3.20iT - 23T^{2} \) |
| 29 | \( 1 + 3.11T + 29T^{2} \) |
| 31 | \( 1 + 6.03T + 31T^{2} \) |
| 37 | \( 1 + 5.49T + 37T^{2} \) |
| 41 | \( 1 - 8.45iT - 41T^{2} \) |
| 43 | \( 1 + 6.30iT - 43T^{2} \) |
| 47 | \( 1 + 1.42T + 47T^{2} \) |
| 53 | \( 1 + 2.54T + 53T^{2} \) |
| 59 | \( 1 - 3.43T + 59T^{2} \) |
| 61 | \( 1 + 1.43iT - 61T^{2} \) |
| 67 | \( 1 + 9.76iT - 67T^{2} \) |
| 71 | \( 1 - 12.9iT - 71T^{2} \) |
| 73 | \( 1 + 1.80iT - 73T^{2} \) |
| 79 | \( 1 - 12.4iT - 79T^{2} \) |
| 83 | \( 1 + 12.2T + 83T^{2} \) |
| 89 | \( 1 + 1.29iT - 89T^{2} \) |
| 97 | \( 1 + 2.88iT - 97T^{2} \) |
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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
−11.02200364167654733719019193463, −10.30940539178190464286397197978, −9.598254066408880251238152493456, −8.352617441833623032003052136558, −7.40021576484757111873355695041, −6.67617263006496330157684949498, −5.60734586929949077497804844576, −3.92552066685628910194868436570, −3.15913184162686523918128217971, −1.84982467219822929809115149879,
0.36970611306705854536781962556, 1.69924975948904748181173571681, 4.28548018452238644423460554341, 5.07082717404925040243190649594, 5.69108262011216895264534819223, 6.92024853999973038519434788490, 7.75902728948600471267691427292, 8.788001961145042549330769018338, 9.289923764109490120524705797392, 10.13986198889455074182426955486