| L(s) = 1 | − 3·3-s + 5.77i·5-s + (3.77 − 18.1i)7-s + 9·9-s − 14.5i·11-s − 75.4i·13-s − 17.3i·15-s − 81.5i·17-s − 89.0·19-s + (−11.3 + 54.3i)21-s + 9.53i·23-s + 91.6·25-s − 27·27-s − 259.·29-s + 13.5·31-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.516i·5-s + (0.203 − 0.978i)7-s + 0.333·9-s − 0.398i·11-s − 1.61i·13-s − 0.298i·15-s − 1.16i·17-s − 1.07·19-s + (−0.117 + 0.565i)21-s + 0.0864i·23-s + 0.733·25-s − 0.192·27-s − 1.66·29-s + 0.0783·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 - 0.203i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.978 - 0.203i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.5062157777\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5062157777\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 7 | \( 1 + (-3.77 + 18.1i)T \) |
| good | 5 | \( 1 - 5.77iT - 125T^{2} \) |
| 11 | \( 1 + 14.5iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 75.4iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 81.5iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 89.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 9.53iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 259.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 13.5T + 2.97e4T^{2} \) |
| 37 | \( 1 - 196.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 341. iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 535. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 26.7T + 1.03e5T^{2} \) |
| 53 | \( 1 - 671.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 546.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 815. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 605. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 287. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 64.3iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 131. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 822.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 737. iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 348. iT - 9.12e5T^{2} \) |
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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
−8.810421951472576076066596833622, −7.70519919031663236186943288661, −7.29950621087679113233661962437, −6.30319079145957552468552563962, −5.50789686538783719254957970671, −4.58856256718779824521452586934, −3.58037400629377431875204814183, −2.60607113828660658102936697003, −1.03009720880132981255845050596, −0.14083672972713213907199090807,
1.53885738590704737302890781068, 2.25022324583287791234085966662, 3.93683355855274907477516864620, 4.60917210993176171641554788508, 5.55181482128226647360040264831, 6.30069149489294702092828956251, 7.07071862470139916250412807254, 8.236574947399971787312338091579, 8.907120673170305754227699265889, 9.506998037767334314389616526455
