| L(s) = 1 | + (2.88 + 1.92i)3-s + (2.22 − 0.441i)5-s + (3.75 + 0.747i)7-s + (1.14 + 2.77i)9-s + (2.61 + 3.90i)11-s + (−0.574 + 0.574i)13-s + (7.24 + 3.00i)15-s + (−11.8 + 12.1i)17-s + (2.04 − 4.94i)19-s + (9.38 + 9.38i)21-s + (15.0 − 10.0i)23-s + (−18.3 + 7.60i)25-s + (4.05 − 20.3i)27-s + (−0.159 − 0.802i)29-s + (11.9 − 17.9i)31-s + ⋯ |
| L(s) = 1 | + (0.960 + 0.641i)3-s + (0.444 − 0.0883i)5-s + (0.536 + 0.106i)7-s + (0.127 + 0.308i)9-s + (0.237 + 0.355i)11-s + (−0.0441 + 0.0441i)13-s + (0.483 + 0.200i)15-s + (−0.696 + 0.717i)17-s + (0.107 − 0.260i)19-s + (0.447 + 0.447i)21-s + (0.653 − 0.436i)23-s + (−0.734 + 0.304i)25-s + (0.150 − 0.754i)27-s + (−0.00550 − 0.0276i)29-s + (0.386 − 0.578i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.847 - 0.530i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.847 - 0.530i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.95905 + 0.562399i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.95905 + 0.562399i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + (11.8 - 12.1i)T \) |
| good | 3 | \( 1 + (-2.88 - 1.92i)T + (3.44 + 8.31i)T^{2} \) |
| 5 | \( 1 + (-2.22 + 0.441i)T + (23.0 - 9.56i)T^{2} \) |
| 7 | \( 1 + (-3.75 - 0.747i)T + (45.2 + 18.7i)T^{2} \) |
| 11 | \( 1 + (-2.61 - 3.90i)T + (-46.3 + 111. i)T^{2} \) |
| 13 | \( 1 + (0.574 - 0.574i)T - 169iT^{2} \) |
| 19 | \( 1 + (-2.04 + 4.94i)T + (-255. - 255. i)T^{2} \) |
| 23 | \( 1 + (-15.0 + 10.0i)T + (202. - 488. i)T^{2} \) |
| 29 | \( 1 + (0.159 + 0.802i)T + (-776. + 321. i)T^{2} \) |
| 31 | \( 1 + (-11.9 + 17.9i)T + (-367. - 887. i)T^{2} \) |
| 37 | \( 1 + (14.4 + 9.62i)T + (523. + 1.26e3i)T^{2} \) |
| 41 | \( 1 + (54.5 + 10.8i)T + (1.55e3 + 643. i)T^{2} \) |
| 43 | \( 1 + (15.8 + 38.3i)T + (-1.30e3 + 1.30e3i)T^{2} \) |
| 47 | \( 1 + (1.14 - 1.14i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (-0.621 + 1.50i)T + (-1.98e3 - 1.98e3i)T^{2} \) |
| 59 | \( 1 + (101. - 42.0i)T + (2.46e3 - 2.46e3i)T^{2} \) |
| 61 | \( 1 + (-16.2 + 81.7i)T + (-3.43e3 - 1.42e3i)T^{2} \) |
| 67 | \( 1 - 3.15iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (-82.9 - 55.4i)T + (1.92e3 + 4.65e3i)T^{2} \) |
| 73 | \( 1 + (-62.2 + 12.3i)T + (4.92e3 - 2.03e3i)T^{2} \) |
| 79 | \( 1 + (-38.9 - 58.2i)T + (-2.38e3 + 5.76e3i)T^{2} \) |
| 83 | \( 1 + (-46.5 - 19.2i)T + (4.87e3 + 4.87e3i)T^{2} \) |
| 89 | \( 1 + (-83.0 - 83.0i)T + 7.92e3iT^{2} \) |
| 97 | \( 1 + (4.35 + 21.8i)T + (-8.69e3 + 3.60e3i)T^{2} \) |
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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
−13.32692462569268580809308896681, −12.07249122360824866783549657966, −10.85813147555579738438667767049, −9.765536916273063657510357810996, −8.956148539131618356327254573171, −8.057787086485114559096364065443, −6.54444264539327585778777698328, −4.97816844265010087578251912607, −3.72171723706721376023465591015, −2.12381363442733920025144378161,
1.72139004180737025060670291059, 3.12532065780670376718061069303, 4.95689048263603889661690322936, 6.52413768419129841184297927089, 7.66765376990212670190303059536, 8.567258474346549407803470303418, 9.562670394565911745359704433277, 10.88734146767915972575309132810, 11.94432275098661057306963393566, 13.26106092471276136694707003545
