| L(s) = 1 | + (0.556 + 5.16i)3-s − 10.6i·5-s − 7.90i·7-s + (−26.3 + 5.75i)9-s + 11.7·11-s − 30.5·13-s + (54.8 − 5.90i)15-s + 118. i·17-s + 66.4i·19-s + (40.8 − 4.40i)21-s + 166.·23-s + 12.4·25-s + (−44.4 − 133. i)27-s + 111. i·29-s + 224. i·31-s + ⋯ |
| L(s) = 1 | + (0.107 + 0.994i)3-s − 0.948i·5-s − 0.426i·7-s + (−0.977 + 0.213i)9-s + 0.322·11-s − 0.652·13-s + (0.943 − 0.101i)15-s + 1.68i·17-s + 0.802i·19-s + (0.424 − 0.0457i)21-s + 1.50·23-s + 0.0998·25-s + (−0.316 − 0.948i)27-s + 0.717i·29-s + 1.30i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.107 - 0.994i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.107 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.534117456\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.534117456\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.556 - 5.16i)T \) |
| good | 5 | \( 1 + 10.6iT - 125T^{2} \) |
| 7 | \( 1 + 7.90iT - 343T^{2} \) |
| 11 | \( 1 - 11.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 30.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 118. iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 66.4iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 166.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 111. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 224. iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 70.6T + 5.06e4T^{2} \) |
| 41 | \( 1 - 247. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 98.7iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 189.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 529. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 811.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 833.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 2.83iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 796.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 875.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 656. iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 237.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 868. iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 755.T + 9.12e5T^{2} \) |
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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
−10.82528727279604619223247604365, −10.36783792848347333753337973433, −9.181477340036554954749466301708, −8.700626187378621250972879155715, −7.61360932691452697519594752832, −6.18488244861326926725743291989, −5.06376755686566594516789876272, −4.31752659920432563447791829892, −3.20194673413285164947073889198, −1.32398102661071219618847517017,
0.55082894371342139991919662256, 2.37409127944431197633157198347, 3.03515570403110466808961616113, 4.88829020313688489553646843843, 6.08784216837240260949058602031, 7.08130794278130670539379488754, 7.48079603923740343450919728443, 8.877395304064619295153280956381, 9.577512309444576085446167810918, 10.99017352629401284934904048204
