| L(s) = 1 | + 2.99i·3-s + 5-s + (−0.183 + 2.63i)7-s − 5.98·9-s − 4.87·11-s − 2.42·13-s + 2.99i·15-s − 3.92i·17-s − 5.24i·19-s + (−7.91 − 0.549i)21-s − 0.114i·23-s + 25-s − 8.94i·27-s + 3.60i·29-s − 4.62·31-s + ⋯ |
| L(s) = 1 | + 1.73i·3-s + 0.447·5-s + (−0.0693 + 0.997i)7-s − 1.99·9-s − 1.47·11-s − 0.673·13-s + 0.773i·15-s − 0.953i·17-s − 1.20i·19-s + (−1.72 − 0.119i)21-s − 0.0237i·23-s + 0.200·25-s − 1.72i·27-s + 0.670i·29-s − 0.831·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.704 + 0.709i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.704 + 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6861927378\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6861927378\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + (0.183 - 2.63i)T \) |
| good | 3 | \( 1 - 2.99iT - 3T^{2} \) |
| 11 | \( 1 + 4.87T + 11T^{2} \) |
| 13 | \( 1 + 2.42T + 13T^{2} \) |
| 17 | \( 1 + 3.92iT - 17T^{2} \) |
| 19 | \( 1 + 5.24iT - 19T^{2} \) |
| 23 | \( 1 + 0.114iT - 23T^{2} \) |
| 29 | \( 1 - 3.60iT - 29T^{2} \) |
| 31 | \( 1 + 4.62T + 31T^{2} \) |
| 37 | \( 1 - 7.83iT - 37T^{2} \) |
| 41 | \( 1 - 10.4iT - 41T^{2} \) |
| 43 | \( 1 + 2.76T + 43T^{2} \) |
| 47 | \( 1 - 12.0T + 47T^{2} \) |
| 53 | \( 1 + 0.668iT - 53T^{2} \) |
| 59 | \( 1 + 1.37iT - 59T^{2} \) |
| 61 | \( 1 + 1.17T + 61T^{2} \) |
| 67 | \( 1 + 2.57T + 67T^{2} \) |
| 71 | \( 1 - 9.43iT - 71T^{2} \) |
| 73 | \( 1 - 5.62iT - 73T^{2} \) |
| 79 | \( 1 - 11.8iT - 79T^{2} \) |
| 83 | \( 1 + 5.52iT - 83T^{2} \) |
| 89 | \( 1 + 6.21iT - 89T^{2} \) |
| 97 | \( 1 + 7.85iT - 97T^{2} \) |
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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
−10.15119724441389543428268393068, −9.601399427342233696163898702699, −8.995268457677721129079830381985, −8.199474107057872774922665299972, −6.97844176400056488212963841539, −5.64027915915605330284379832349, −5.15175589698820644816399000947, −4.56335949798990200646267420223, −3.00017696402697846279857390538, −2.59769195159984313928929973830,
0.27682548994297917167094758350, 1.69824052205089232805528765693, 2.50767469748326817843360150022, 3.86106993138929408919765206336, 5.38422912821586123466905831903, 5.99962637429715256304272191165, 7.00348196634548522132431168272, 7.65889191797028565561385822412, 8.045678491015149240984897948751, 9.207055163310802099165055151098
