| L(s) = 1 | − 6i·5-s − 21.1·7-s + 42.3i·11-s − 20i·13-s + 8·17-s + 84.6i·19-s + 169.·23-s + 89·25-s − 46i·29-s + 21.1·31-s + 126. i·35-s − 164i·37-s − 312·41-s − 423. i·43-s − 169.·47-s + ⋯ |
| L(s) = 1 | − 0.536i·5-s − 1.14·7-s + 1.16i·11-s − 0.426i·13-s + 0.114·17-s + 1.02i·19-s + 1.53·23-s + 0.711·25-s − 0.294i·29-s + 0.122·31-s + 0.613i·35-s − 0.728i·37-s − 1.18·41-s − 1.50i·43-s − 0.525·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6691229540\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6691229540\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 6iT - 125T^{2} \) |
| 7 | \( 1 + 21.1T + 343T^{2} \) |
| 11 | \( 1 - 42.3iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 20iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 84.6iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 169.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 46iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 21.1T + 2.97e4T^{2} \) |
| 37 | \( 1 + 164iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 312T + 6.89e4T^{2} \) |
| 43 | \( 1 + 423. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 169.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 266iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 253. iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 132iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 507. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 677.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 246T + 3.89e5T^{2} \) |
| 79 | \( 1 + 232.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 973. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 1.39e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 302T + 9.12e5T^{2} \) |
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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
−9.155976428761821389169852970025, −8.423770315352353201425632992008, −7.30918388910267536820033015636, −6.73447169633712631361350903956, −5.65347616855220775776091121568, −4.85134172251848842998253374235, −3.77781344438274703600685602465, −2.82864035433359919397687291548, −1.50701856831571009167156852478, −0.17824465339069045753305772108,
1.10105881699621432751345220702, 2.96907258457680943611021892785, 3.11875767581333603192759740542, 4.54520414980913956935308001048, 5.59942407148581592771416956372, 6.70452409333778366591594433586, 6.82726479786059202067521158133, 8.180281603052474915530171895011, 9.028766306522555490089514479270, 9.612584530081858222026007341083
