| L(s) = 1 | + (1.86 + 5.72i)3-s + (−61.9 − 45.0i)5-s + (−34.8 + 107. i)7-s + (167. − 121. i)9-s + (361. + 174. i)11-s + (475. − 345. i)13-s + (142. − 438. i)15-s + (611. + 444. i)17-s + (249. + 767. i)19-s − 678.·21-s + 2.02e3·23-s + (848. + 2.61e3i)25-s + (2.19e3 + 1.59e3i)27-s + (−966. + 2.97e3i)29-s + (7.15e3 − 5.20e3i)31-s + ⋯ |
| L(s) = 1 | + (0.119 + 0.367i)3-s + (−1.10 − 0.805i)5-s + (−0.268 + 0.827i)7-s + (0.688 − 0.500i)9-s + (0.900 + 0.434i)11-s + (0.779 − 0.566i)13-s + (0.163 − 0.503i)15-s + (0.513 + 0.373i)17-s + (0.158 + 0.487i)19-s − 0.335·21-s + 0.799·23-s + (0.271 + 0.835i)25-s + (0.578 + 0.420i)27-s + (−0.213 + 0.656i)29-s + (1.33 − 0.971i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.154i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.987 - 0.154i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.69621 + 0.131807i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.69621 + 0.131807i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + (-361. - 174. i)T \) |
| good | 3 | \( 1 + (-1.86 - 5.72i)T + (-196. + 142. i)T^{2} \) |
| 5 | \( 1 + (61.9 + 45.0i)T + (965. + 2.97e3i)T^{2} \) |
| 7 | \( 1 + (34.8 - 107. i)T + (-1.35e4 - 9.87e3i)T^{2} \) |
| 13 | \( 1 + (-475. + 345. i)T + (1.14e5 - 3.53e5i)T^{2} \) |
| 17 | \( 1 + (-611. - 444. i)T + (4.38e5 + 1.35e6i)T^{2} \) |
| 19 | \( 1 + (-249. - 767. i)T + (-2.00e6 + 1.45e6i)T^{2} \) |
| 23 | \( 1 - 2.02e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + (966. - 2.97e3i)T + (-1.65e7 - 1.20e7i)T^{2} \) |
| 31 | \( 1 + (-7.15e3 + 5.20e3i)T + (8.84e6 - 2.72e7i)T^{2} \) |
| 37 | \( 1 + (-4.81e3 + 1.48e4i)T + (-5.61e7 - 4.07e7i)T^{2} \) |
| 41 | \( 1 + (-3.99e3 - 1.23e4i)T + (-9.37e7 + 6.80e7i)T^{2} \) |
| 43 | \( 1 + 7.18e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-3.22e3 - 9.92e3i)T + (-1.85e8 + 1.34e8i)T^{2} \) |
| 53 | \( 1 + (2.46e4 - 1.78e4i)T + (1.29e8 - 3.97e8i)T^{2} \) |
| 59 | \( 1 + (-9.18e3 + 2.82e4i)T + (-5.78e8 - 4.20e8i)T^{2} \) |
| 61 | \( 1 + (2.51e4 + 1.82e4i)T + (2.60e8 + 8.03e8i)T^{2} \) |
| 67 | \( 1 - 5.56e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (3.66e4 + 2.66e4i)T + (5.57e8 + 1.71e9i)T^{2} \) |
| 73 | \( 1 + (4.22e3 - 1.30e4i)T + (-1.67e9 - 1.21e9i)T^{2} \) |
| 79 | \( 1 + (1.23e4 - 8.98e3i)T + (9.50e8 - 2.92e9i)T^{2} \) |
| 83 | \( 1 + (3.25e4 + 2.36e4i)T + (1.21e9 + 3.74e9i)T^{2} \) |
| 89 | \( 1 + 8.25e3T + 5.58e9T^{2} \) |
| 97 | \( 1 + (3.69e4 - 2.68e4i)T + (2.65e9 - 8.16e9i)T^{2} \) |
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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
−12.78777418769507058332346800569, −12.33650482691677076817635125007, −11.23058250249438667045338434756, −9.689521646839313577697034079247, −8.826328679442123120933505350996, −7.72248327584911711173390914905, −6.13995327096993321013652624365, −4.53310265767466364777143092961, −3.48689381403982057187415021173, −1.05264126714320354655126388165,
1.02012296446749987245482139306, 3.25300726422544877643126079847, 4.36605667768704602570556367729, 6.63860887752500863482281614696, 7.26718679033342873752991070471, 8.460765761783235073935326531799, 10.04782828940788957774194902433, 11.14002784150784330830635486408, 11.90700589907046323387723110063, 13.38056743001441244108924912741
