| L(s) = 1 | + (−1.06 − 1.32i)2-s + (−0.198 + 0.868i)4-s + (1.45 + 0.993i)5-s + (−0.297 + 0.514i)7-s + (−1.69 + 0.818i)8-s + (−0.224 − 2.99i)10-s + (−0.967 − 4.23i)11-s + (0.242 − 3.23i)13-s + (0.999 − 0.150i)14-s + (4.49 + 2.16i)16-s + (4.57 − 3.12i)17-s + (2.59 − 0.801i)19-s + (−1.15 + 1.06i)20-s + (−4.60 + 5.77i)22-s + (3.88 − 3.60i)23-s + ⋯ |
| L(s) = 1 | + (−0.749 − 0.940i)2-s + (−0.0991 + 0.434i)4-s + (0.651 + 0.444i)5-s + (−0.112 + 0.194i)7-s + (−0.600 + 0.289i)8-s + (−0.0708 − 0.945i)10-s + (−0.291 − 1.27i)11-s + (0.0672 − 0.896i)13-s + (0.267 − 0.0402i)14-s + (1.12 + 0.541i)16-s + (1.11 − 0.757i)17-s + (0.595 − 0.183i)19-s + (−0.257 + 0.239i)20-s + (−0.982 + 1.23i)22-s + (0.809 − 0.751i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.459 + 0.888i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.459 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.487400 - 0.801045i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.487400 - 0.801045i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 43 | \( 1 + (4.98 + 4.26i)T \) |
| good | 2 | \( 1 + (1.06 + 1.32i)T + (-0.445 + 1.94i)T^{2} \) |
| 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
−10.93107173351391325627836416379, −10.15272231649766783378044180479, −9.452941365313701530257775152264, −8.557497301097796015110400122113, −7.52955272643616871237380071158, −6.01225896226334307637354559507, −5.44761362617145052632534011637, −3.28925976091003799339479812416, −2.57724474329194547751133184007, −0.846411456882905809837853054113,
1.66592377968938298846834853218, 3.59811260905357940484750371231, 5.14508085576556250753750142112, 6.03267044641468080120235734194, 7.21841371945644567449469466566, 7.67736560730084248838495202391, 9.000368091543182061269112084201, 9.502939085297841690809279852144, 10.26609506349573236699619444068, 11.67752689856176269856676162522
