| L(s) = 1 | − 1.63i·3-s + (−2.23 − 0.153i)5-s + i·7-s + 0.328·9-s − 1.24·11-s + 4.20i·13-s + (−0.250 + 3.64i)15-s + 3.39i·17-s + 6.46·19-s + 1.63·21-s + 2.15i·23-s + (4.95 + 0.683i)25-s − 5.44i·27-s − 3.96·29-s + 10.0·31-s + ⋯ |
| L(s) = 1 | − 0.943i·3-s + (−0.997 − 0.0685i)5-s + 0.377i·7-s + 0.109·9-s − 0.374·11-s + 1.16i·13-s + (−0.0646 + 0.941i)15-s + 0.823i·17-s + 1.48·19-s + 0.356·21-s + 0.449i·23-s + (0.990 + 0.136i)25-s − 1.04i·27-s − 0.736·29-s + 1.80·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0685i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0685i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.333536921\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.333536921\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (2.23 + 0.153i)T \) |
| 7 | \( 1 - iT \) |
| good | 3 | \( 1 + 1.63iT - 3T^{2} \) |
| 11 | \( 1 + 1.24T + 11T^{2} \) |
| 13 | \( 1 - 4.20iT - 13T^{2} \) |
| 17 | \( 1 - 3.39iT - 17T^{2} \) |
| 19 | \( 1 - 6.46T + 19T^{2} \) |
| 23 | \( 1 - 2.15iT - 23T^{2} \) |
| 29 | \( 1 + 3.96T + 29T^{2} \) |
| 31 | \( 1 - 10.0T + 31T^{2} \) |
| 37 | \( 1 + 6.76iT - 37T^{2} \) |
| 41 | \( 1 + 0.131T + 41T^{2} \) |
| 43 | \( 1 + 7.40iT - 43T^{2} \) |
| 47 | \( 1 - 4.82iT - 47T^{2} \) |
| 53 | \( 1 - 10.0iT - 53T^{2} \) |
| 59 | \( 1 - 10.9T + 59T^{2} \) |
| 61 | \( 1 + 6.33T + 61T^{2} \) |
| 67 | \( 1 - 2.65iT - 67T^{2} \) |
| 71 | \( 1 + 0.754T + 71T^{2} \) |
| 73 | \( 1 - 6.03iT - 73T^{2} \) |
| 79 | \( 1 - 14.6T + 79T^{2} \) |
| 83 | \( 1 - 14.0iT - 83T^{2} \) |
| 89 | \( 1 - 12.6T + 89T^{2} \) |
| 97 | \( 1 + 0.914iT - 97T^{2} \) |
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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
−9.719083030311344402702618136971, −8.861615607491800256126106358911, −7.964311054725694117851497855703, −7.41045137809122897598605450430, −6.69203538277729928321727426175, −5.68375639826399958964507200149, −4.53954529580680138688536212385, −3.61876102259087331397881670833, −2.31768480279854528430991147506, −1.08046439096713055611024755420,
0.76051904040828307288564385906, 2.96693950208420419442696808021, 3.58542445427792752040607136376, 4.72280397031164105487027960232, 5.15879105180176691363170630157, 6.58681468923159314117382433071, 7.58799012780162969530735674772, 8.039554746915273597197866426081, 9.144979207420570175852871518402, 10.05627704733567137109369978014
