| L(s) = 1 | + 12.3i·5-s + 21.4i·7-s − 15.5·11-s − 2·13-s + 24.7i·17-s + 85.6i·19-s − 124.·23-s − 27.9·25-s − 272. i·29-s − 278. i·31-s − 265.·35-s + 128·37-s + 296. i·41-s − 42.8i·43-s − 592.·47-s + ⋯ |
| L(s) = 1 | + 1.10i·5-s + 1.15i·7-s − 0.427·11-s − 0.0426·13-s + 0.352i·17-s + 1.03i·19-s − 1.13·23-s − 0.223·25-s − 1.74i·29-s − 1.61i·31-s − 1.27·35-s + 0.568·37-s + 1.13i·41-s − 0.151i·43-s − 1.83·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8914841334\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8914841334\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 12.3iT - 125T^{2} \) |
| 7 | \( 1 - 21.4iT - 343T^{2} \) |
| 11 | \( 1 + 15.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 24.7iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 85.6iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 124.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 272. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 278. iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 128T + 5.06e4T^{2} \) |
| 41 | \( 1 - 296. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 42.8iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 592.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 259. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 530.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 340T + 2.26e5T^{2} \) |
| 67 | \( 1 - 899. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 966.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 817T + 3.89e5T^{2} \) |
| 79 | \( 1 - 214. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 358.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 915. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 965T + 9.12e5T^{2} \) |
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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
−11.29010122376503711494913806426, −10.18263750497562831447497832589, −9.605597375827719697380407576055, −8.271439012810719172821251288729, −7.65787693277461397883305921115, −6.22002079000557253059737668710, −5.85142592237865463173727599084, −4.27620730882754556140098380650, −2.96563156863142463859765261970, −2.06154878611918138817789955728,
0.28472710602980690226419601398, 1.49347763875534544180407815435, 3.27474073236677394853512132858, 4.56293349876928932537543826528, 5.17523392849378332063202812408, 6.64403406343019837191535275126, 7.53019044534645123921506682827, 8.505649414406756092502055260485, 9.313448174070513464642292375040, 10.34437640417272867410988132201
