| L(s) = 1 | − 3i·3-s − 5.67i·5-s + 33.0·7-s − 9·9-s + 34.6i·11-s + 82.2i·13-s − 17.0·15-s + 97.8·17-s + 55.8i·19-s − 99.2i·21-s − 130.·23-s + 92.7·25-s + 27i·27-s − 147. i·29-s − 101.·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 0.507i·5-s + 1.78·7-s − 0.333·9-s + 0.949i·11-s + 1.75i·13-s − 0.293·15-s + 1.39·17-s + 0.674i·19-s − 1.03i·21-s − 1.18·23-s + 0.742·25-s + 0.192i·27-s − 0.944i·29-s − 0.586·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.342845514\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.342845514\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3iT \) |
| good | 5 | \( 1 + 5.67iT - 125T^{2} \) |
| 7 | \( 1 - 33.0T + 343T^{2} \) |
| 11 | \( 1 - 34.6iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 82.2iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 97.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 55.8iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 130.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 147. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 101.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 184. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 237.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 199. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 334.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 102. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 105. iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 717. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 316. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 800.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 301.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 42.8T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.23e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 1.32e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 505.T + 9.12e5T^{2} \) |
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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
−11.15442584366350690946693824957, −9.953122371476158641052485191612, −8.941918527584486058874165456033, −7.957049846229254429169169749343, −7.44070302337166028436326400359, −6.09584053176418991870089823380, −4.93023411770112291767569032272, −4.16211174296075620475750147891, −2.03917448723456304114599333653, −1.34427720714433291050040697241,
0.942284741161076249285924439161, 2.70273725952707378567414619786, 3.83260629071676326944981521178, 5.23827461800257500343042988104, 5.69409115542563174472255821506, 7.45686821276609728758520869908, 8.102979100467464860501742148540, 8.946960106670096525422693897338, 10.45732231449068863559435062360, 10.68105211999744189783558151794
