| L(s) = 1 | + (−1.44 + 0.612i)2-s + (0.991 − 1.42i)3-s + (0.325 − 0.335i)4-s + (−3.30 + 2.33i)5-s + (−0.564 + 2.66i)6-s + (−4.08 + 1.58i)7-s + (0.869 − 2.24i)8-s + (−1.03 − 2.81i)9-s + (3.34 − 5.40i)10-s + (−2.63 + 3.48i)11-s + (−0.153 − 0.794i)12-s + (−0.654 + 4.21i)13-s + (4.94 − 4.79i)14-s + (0.0454 + 7.01i)15-s + (0.145 + 4.71i)16-s + (−2.59 − 0.914i)17-s + ⋯ |
| L(s) = 1 | + (−1.02 + 0.432i)2-s + (0.572 − 0.819i)3-s + (0.162 − 0.167i)4-s + (−1.47 + 1.04i)5-s + (−0.230 + 1.08i)6-s + (−1.54 + 0.598i)7-s + (0.307 − 0.793i)8-s + (−0.344 − 0.938i)9-s + (1.05 − 1.70i)10-s + (−0.793 + 1.05i)11-s + (−0.0444 − 0.229i)12-s + (−0.181 + 1.16i)13-s + (1.32 − 1.28i)14-s + (0.0117 + 1.81i)15-s + (0.0363 + 1.17i)16-s + (−0.629 − 0.221i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.796i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 + 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.185644 - 0.0920686i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.185644 - 0.0920686i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.991 + 1.42i)T \) |
| 103 | \( 1 + (9.97 - 1.86i)T \) |
| good | 2 | \( 1 + (1.44 - 0.612i)T + (1.39 - 1.43i)T^{2} \) |
| 5 | \( 1 + (3.30 - 2.33i)T + (1.66 - 4.71i)T^{2} \) |
| 7 | \( 1 + (4.08 - 1.58i)T + (5.17 - 4.71i)T^{2} \) |
| 11 | \( 1 + (2.63 - 3.48i)T + (-3.01 - 10.5i)T^{2} \) |
| 13 | \( 1 + (0.654 - 4.21i)T + (-12.3 - 3.94i)T^{2} \) |
| 17 | \( 1 + (2.59 + 0.914i)T + (13.2 + 10.6i)T^{2} \) |
| 19 | \( 1 + (-6.49 + 0.400i)T + (18.8 - 2.33i)T^{2} \) |
| 23 | \( 1 + (0.606 + 4.89i)T + (-22.3 + 5.61i)T^{2} \) |
| 29 | \( 1 + (-0.997 + 2.16i)T + (-18.8 - 22.0i)T^{2} \) |
| 31 | \( 1 + (-1.94 + 0.0600i)T + (30.9 - 1.90i)T^{2} \) |
| 37 | \( 1 + (-2.94 - 3.23i)T + (-3.41 + 36.8i)T^{2} \) |
| 41 | \( 1 + (4.77 + 0.442i)T + (40.3 + 7.53i)T^{2} \) |
| 43 | \( 1 + (2.72 + 8.57i)T + (-35.0 + 24.8i)T^{2} \) |
| 47 | \( 1 + 0.328T + 47T^{2} \) |
| 53 | \( 1 + (-0.214 - 0.323i)T + (-20.6 + 48.8i)T^{2} \) |
| 59 | \( 1 + (4.52 - 11.6i)T + (-43.6 - 39.7i)T^{2} \) |
| 61 | \( 1 + (-4.75 - 5.55i)T + (-9.35 + 60.2i)T^{2} \) |
| 67 | \( 1 + (-0.0584 + 0.00907i)T + (63.8 - 20.3i)T^{2} \) |
| 71 | \( 1 + (2.66 + 1.88i)T + (23.5 + 66.9i)T^{2} \) |
| 73 | \( 1 + (1.68 + 0.155i)T + (71.7 + 13.4i)T^{2} \) |
| 79 | \( 1 + (-9.33 + 6.60i)T + (26.2 - 74.5i)T^{2} \) |
| 83 | \( 1 + (4.13 + 0.641i)T + (79.0 + 25.1i)T^{2} \) |
| 89 | \( 1 + (-3.04 + 10.6i)T + (-75.6 - 46.8i)T^{2} \) |
| 97 | \( 1 + (6.04 - 7.06i)T + (-14.8 - 95.8i)T^{2} \) |
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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
−9.712923924668055859189710933225, −8.985156388490328301536841387733, −8.182034209705867202720272871240, −7.26219315520335134937368352492, −7.03497844924877475072157327305, −6.35133798125970070881040190055, −4.32130334024889362998627535101, −3.29631421129438627316701725398, −2.47521542660886633512842286666, −0.19835569016293798447068723881,
0.76676429923712869632213437230, 3.09296767100129067312557342341, 3.50786104319521794992512477816, 4.79485112277459735751540931640, 5.58663443784544857910657282373, 7.41242923494293357781375882142, 8.057578479835146582021180294990, 8.537355275236917271744486389047, 9.543894663904927526361509803737, 9.862073183961349282196313049138
