| L(s) = 1 | + (−1.26 + 0.633i)2-s + (2.28 + 0.895i)3-s + (1.19 − 1.60i)4-s + (2.72 − 0.204i)5-s + (−3.45 + 0.312i)6-s + (−0.893 + 1.54i)7-s + (−0.499 + 2.78i)8-s + (2.20 + 2.04i)9-s + (−3.31 + 1.98i)10-s + (−3.16 − 0.722i)11-s + (4.16 − 2.58i)12-s + (2.99 + 4.39i)13-s + (0.149 − 2.52i)14-s + (6.39 + 1.97i)15-s + (−1.13 − 3.83i)16-s + (−0.00441 + 0.0588i)17-s + ⋯ |
| L(s) = 1 | + (−0.894 + 0.447i)2-s + (1.31 + 0.516i)3-s + (0.598 − 0.800i)4-s + (1.21 − 0.0912i)5-s + (−1.40 + 0.127i)6-s + (−0.337 + 0.584i)7-s + (−0.176 + 0.984i)8-s + (0.733 + 0.680i)9-s + (−1.04 + 0.626i)10-s + (−0.954 − 0.217i)11-s + (1.20 − 0.745i)12-s + (0.831 + 1.21i)13-s + (0.0399 − 0.674i)14-s + (1.65 + 0.508i)15-s + (−0.282 − 0.959i)16-s + (−0.00106 + 0.0142i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.503 - 0.863i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.503 - 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.32995 + 0.763820i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.32995 + 0.763820i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.26 - 0.633i)T \) |
| 43 | \( 1 + (-3.42 - 5.59i)T \) |
| good | 3 | \( 1 + (-2.28 - 0.895i)T + (2.19 + 2.04i)T^{2} \) |
| 5 | \( 1 + (-2.72 + 0.204i)T + (4.94 - 0.745i)T^{2} \) |
| 7 | \( 1 + (0.893 - 1.54i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (3.16 + 0.722i)T + (9.91 + 4.77i)T^{2} \) |
| 13 | \( 1 + (-2.99 - 4.39i)T + (-4.74 + 12.1i)T^{2} \) |
| 17 | \( 1 + (0.00441 - 0.0588i)T + (-16.8 - 2.53i)T^{2} \) |
| 19 | \( 1 + (4.92 + 5.30i)T + (-1.41 + 18.9i)T^{2} \) |
| 23 | \( 1 + (-9.09 + 2.80i)T + (19.0 - 12.9i)T^{2} \) |
| 29 | \( 1 + (-1.16 + 0.458i)T + (21.2 - 19.7i)T^{2} \) |
| 31 | \( 1 + (4.44 + 0.670i)T + (29.6 + 9.13i)T^{2} \) |
| 37 | \( 1 + (4.92 - 2.84i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.62 + 2.03i)T + (-9.12 - 39.9i)T^{2} \) |
| 47 | \( 1 + (2.35 + 10.2i)T + (-42.3 + 20.3i)T^{2} \) |
| 53 | \( 1 + (3.92 - 5.76i)T + (-19.3 - 49.3i)T^{2} \) |
| 59 | \( 1 + (1.51 + 3.14i)T + (-36.7 + 46.1i)T^{2} \) |
| 61 | \( 1 + (0.727 + 4.82i)T + (-58.2 + 17.9i)T^{2} \) |
| 67 | \( 1 + (6.03 + 6.49i)T + (-5.00 + 66.8i)T^{2} \) |
| 71 | \( 1 + (-7.25 - 2.23i)T + (58.6 + 39.9i)T^{2} \) |
| 73 | \( 1 + (-0.179 + 0.122i)T + (26.6 - 67.9i)T^{2} \) |
| 79 | \( 1 + (3.48 - 6.02i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (9.89 + 3.88i)T + (60.8 + 56.4i)T^{2} \) |
| 89 | \( 1 + (-5.37 + 13.7i)T + (-65.2 - 60.5i)T^{2} \) |
| 97 | \( 1 + (1.67 - 7.31i)T + (-87.3 - 42.0i)T^{2} \) |
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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
−11.16483405939146414253609179431, −10.42656006340434067963676152061, −9.324578396103919691464683527252, −9.041327015536538938277180377361, −8.395291692023113045480196009702, −6.96367910991092556760754594256, −6.05501056821396256067217706233, −4.83285873489004449226579161190, −2.88500586644836759927960579687, −2.02297464033747303787260338849,
1.50564972499030729794091540065, 2.64235599680328125741661918995, 3.53287418902811619369587466711, 5.71729094714372029463017624727, 6.97664191266334143188723267548, 7.82677581810518801193097843918, 8.621357661399573063295989941509, 9.436538531195308807000346044683, 10.38528504114019997152452299875, 10.81039564998334989172131556558
