| L(s) = 1 | + (−0.846 + 0.532i)2-s + (−0.0459 − 0.0288i)3-s + (0.433 − 0.900i)4-s + (−1.90 + 1.17i)5-s + 0.0542·6-s + (−0.499 − 0.499i)7-s + (0.111 + 0.993i)8-s + (−1.30 − 2.70i)9-s + (0.984 − 2.00i)10-s + (1.98 − 0.956i)11-s + (−0.0459 + 0.0288i)12-s + (4.43 − 0.499i)13-s + (0.688 + 0.157i)14-s + (0.121 + 0.000867i)15-s + (−0.623 − 0.781i)16-s + (2.69 + 0.303i)17-s + ⋯ |
| L(s) = 1 | + (−0.598 + 0.376i)2-s + (−0.0265 − 0.0166i)3-s + (0.216 − 0.450i)4-s + (−0.850 + 0.525i)5-s + 0.0221·6-s + (−0.188 − 0.188i)7-s + (0.0395 + 0.351i)8-s + (−0.433 − 0.900i)9-s + (0.311 − 0.634i)10-s + (0.598 − 0.288i)11-s + (−0.0132 + 0.00833i)12-s + (1.22 − 0.138i)13-s + (0.184 + 0.0420i)14-s + (0.0313 + 0.000224i)15-s + (−0.155 − 0.195i)16-s + (0.652 + 0.0735i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.191i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.981 + 0.191i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.882538 - 0.0855014i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.882538 - 0.0855014i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.846 - 0.532i)T \) |
| 5 | \( 1 + (1.90 - 1.17i)T \) |
| 43 | \( 1 + (-6.51 - 0.709i)T \) |
| good | 3 | \( 1 + (0.0459 + 0.0288i)T + (1.30 + 2.70i)T^{2} \) |
| 7 | \( 1 + (0.499 + 0.499i)T + 7iT^{2} \) |
| 11 | \( 1 + (-1.98 + 0.956i)T + (6.85 - 8.60i)T^{2} \) |
| 13 | \( 1 + (-4.43 + 0.499i)T + (12.6 - 2.89i)T^{2} \) |
| 17 | \( 1 + (-2.69 - 0.303i)T + (16.5 + 3.78i)T^{2} \) |
| 19 | \( 1 + (0.517 + 0.249i)T + (11.8 + 14.8i)T^{2} \) |
| 23 | \( 1 + (-5.75 - 2.01i)T + (17.9 + 14.3i)T^{2} \) |
| 29 | \( 1 + (-0.0156 + 0.0687i)T + (-26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + (-1.84 + 8.08i)T + (-27.9 - 13.4i)T^{2} \) |
| 37 | \( 1 + (1.14 + 1.14i)T + 37iT^{2} \) |
| 41 | \( 1 + (0.823 - 3.60i)T + (-36.9 - 17.7i)T^{2} \) |
| 47 | \( 1 + (-2.75 + 0.964i)T + (36.7 - 29.3i)T^{2} \) |
| 53 | \( 1 + (0.271 - 2.40i)T + (-51.6 - 11.7i)T^{2} \) |
| 59 | \( 1 + (-3.76 + 3.00i)T + (13.1 - 57.5i)T^{2} \) |
| 61 | \( 1 + (-2.12 + 0.484i)T + (54.9 - 26.4i)T^{2} \) |
| 67 | \( 1 + (4.11 - 1.43i)T + (52.3 - 41.7i)T^{2} \) |
| 71 | \( 1 + (4.06 - 8.44i)T + (-44.2 - 55.5i)T^{2} \) |
| 73 | \( 1 + (0.924 + 8.20i)T + (-71.1 + 16.2i)T^{2} \) |
| 79 | \( 1 + 1.84iT - 79T^{2} \) |
| 83 | \( 1 + (4.67 - 7.44i)T + (-36.0 - 74.7i)T^{2} \) |
| 89 | \( 1 + (3.89 + 17.0i)T + (-80.1 + 38.6i)T^{2} \) |
| 97 | \( 1 + (-4.59 + 13.1i)T + (-75.8 - 60.4i)T^{2} \) |
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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
−11.25755190195433108029018737377, −10.20246112294614098485201776234, −9.142465001388492191970701799398, −8.442225258128104797550754130329, −7.44744297981958986772533442402, −6.54818965765930370190197400748, −5.78725293785720894986400102888, −4.02836692664097822846541298683, −3.13012764423745958859694116693, −0.859317978269064585891426265387,
1.25417171289890124181468029045, 3.00771291065763756266818839335, 4.15916579540140516629658867965, 5.35470970188903307836259197351, 6.72959391713076196903484267114, 7.75112530744301734056733558243, 8.626636220637484283529360320819, 9.118614598678220527558657770736, 10.49392666823557432538445956741, 11.09621021446437827742983175086
