| L(s) = 1 | + (1.14 − 0.172i)3-s + (−1.45 + 0.993i)5-s + (0.297 + 0.514i)7-s + (−1.59 + 0.490i)9-s + (−0.967 + 4.23i)11-s + (0.242 + 3.23i)13-s + (−1.49 + 1.38i)15-s + (−4.57 − 3.12i)17-s + (−2.59 − 0.801i)19-s + (0.428 + 0.537i)21-s + (3.88 + 3.60i)23-s + (−0.690 + 1.75i)25-s + (−4.85 + 2.33i)27-s + (9.75 + 1.46i)29-s + (−1.36 − 3.47i)31-s + ⋯ |
| L(s) = 1 | + (0.659 − 0.0994i)3-s + (−0.651 + 0.444i)5-s + (0.112 + 0.194i)7-s + (−0.530 + 0.163i)9-s + (−0.291 + 1.27i)11-s + (0.0672 + 0.896i)13-s + (−0.385 + 0.357i)15-s + (−1.11 − 0.757i)17-s + (−0.595 − 0.183i)19-s + (0.0935 + 0.117i)21-s + (0.809 + 0.751i)23-s + (−0.138 + 0.351i)25-s + (−0.934 + 0.450i)27-s + (1.81 + 0.272i)29-s + (−0.244 − 0.623i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.322 - 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.322 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.668109 + 0.933056i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.668109 + 0.933056i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 43 | \( 1 + (-4.98 + 4.26i)T \) |
| good | 3 | \( 1 + (-1.14 + 0.172i)T + (2.86 - 0.884i)T^{2} \) |
| 5 | \( 1 + (1.45 - 0.993i)T + (1.82 - 4.65i)T^{2} \) |
| 7 | \( 1 + (-0.297 - 0.514i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (0.967 - 4.23i)T + (-9.91 - 4.77i)T^{2} \) |
| 13 | \( 1 + (-0.242 - 3.23i)T + (-12.8 + 1.93i)T^{2} \) |
| 17 | \( 1 + (4.57 + 3.12i)T + (6.21 + 15.8i)T^{2} \) |
| 19 | \( 1 + (2.59 + 0.801i)T + (15.6 + 10.7i)T^{2} \) |
| 23 | \( 1 + (-3.88 - 3.60i)T + (1.71 + 22.9i)T^{2} \) |
| 29 | \( 1 + (-9.75 - 1.46i)T + (27.7 + 8.54i)T^{2} \) |
| 31 | \( 1 + (1.36 + 3.47i)T + (-22.7 + 21.0i)T^{2} \) |
| 37 | \( 1 + (0.673 - 1.16i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (6.99 - 8.76i)T + (-9.12 - 39.9i)T^{2} \) |
| 47 | \( 1 + (-0.909 - 3.98i)T + (-42.3 + 20.3i)T^{2} \) |
| 53 | \( 1 + (0.180 - 2.40i)T + (-52.4 - 7.89i)T^{2} \) |
| 59 | \( 1 + (0.662 - 0.318i)T + (36.7 - 46.1i)T^{2} \) |
| 61 | \( 1 + (-1.86 + 4.76i)T + (-44.7 - 41.4i)T^{2} \) |
| 67 | \( 1 + (-3.67 - 1.13i)T + (55.3 + 37.7i)T^{2} \) |
| 71 | \( 1 + (2.81 - 2.61i)T + (5.30 - 70.8i)T^{2} \) |
| 73 | \( 1 + (0.557 + 7.43i)T + (-72.1 + 10.8i)T^{2} \) |
| 79 | \( 1 + (-6.69 - 11.5i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-7.70 + 1.16i)T + (79.3 - 24.4i)T^{2} \) |
| 89 | \( 1 + (4.45 - 0.672i)T + (85.0 - 26.2i)T^{2} \) |
| 97 | \( 1 + (0.0411 - 0.180i)T + (-87.3 - 42.0i)T^{2} \) |
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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
−10.89150998664907788280215812777, −9.685970388201689524949947496131, −8.977955843848936699849675561597, −8.153021803004452535652301554272, −7.23169767168722786196979163194, −6.63881105557178360417129453388, −5.09623805522268304490393272119, −4.23752723937929358939156764879, −2.96784580553764293867935008444, −2.05967086555247816366077731319,
0.53655241965342491756095537888, 2.56947182024083464747092462068, 3.53154446162449867560893307757, 4.51471532912316553796078872386, 5.71176353244504218643464426341, 6.66124341324781558737178254749, 8.037229739522504117976610543829, 8.474808720758520899574694951622, 8.927135028608471216260056681567, 10.45214957913406567638618043426
