| L(s) = 1 | + (0.766 − 0.642i)2-s + (−0.317 − 0.115i)3-s + (0.173 − 0.984i)4-s + (−0.456 − 2.58i)5-s + (−0.317 + 0.115i)6-s + (0.0977 − 0.169i)7-s + (−0.500 − 0.866i)8-s + (−2.21 − 1.85i)9-s + (−2.01 − 1.68i)10-s + (0.5 + 0.866i)11-s + (−0.168 + 0.292i)12-s + (−2.79 + 1.01i)13-s + (−0.0339 − 0.192i)14-s + (−0.153 + 0.873i)15-s + (−0.939 − 0.342i)16-s + (0.323 − 0.271i)17-s + ⋯ |
| L(s) = 1 | + (0.541 − 0.454i)2-s + (−0.183 − 0.0666i)3-s + (0.0868 − 0.492i)4-s + (−0.204 − 1.15i)5-s + (−0.129 + 0.0471i)6-s + (0.0369 − 0.0640i)7-s + (−0.176 − 0.306i)8-s + (−0.736 − 0.618i)9-s + (−0.636 − 0.534i)10-s + (0.150 + 0.261i)11-s + (−0.0487 + 0.0843i)12-s + (−0.774 + 0.281i)13-s + (−0.00907 − 0.0514i)14-s + (−0.0397 + 0.225i)15-s + (−0.234 − 0.0855i)16-s + (0.0784 − 0.0658i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.531076 - 1.28229i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.531076 - 1.28229i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.766 + 0.642i)T \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
| 19 | \( 1 + (-1.78 + 3.97i)T \) |
| good | 3 | \( 1 + (0.317 + 0.115i)T + (2.29 + 1.92i)T^{2} \) |
| 5 | \( 1 + (0.456 + 2.58i)T + (-4.69 + 1.71i)T^{2} \) |
| 7 | \( 1 + (-0.0977 + 0.169i)T + (-3.5 - 6.06i)T^{2} \) |
| 13 | \( 1 + (2.79 - 1.01i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-0.323 + 0.271i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (0.286 - 1.62i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (2.12 + 1.78i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-3.97 + 6.88i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 9.69T + 37T^{2} \) |
| 41 | \( 1 + (5.95 + 2.16i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (0.0496 + 0.281i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-1.52 - 1.28i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (-0.593 + 3.36i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (-10.8 + 9.12i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (1.45 - 8.27i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-5.15 - 4.32i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-0.300 - 1.70i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-12.7 - 4.63i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (6.92 + 2.51i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (3.51 - 6.09i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (9.28 - 3.38i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-7.68 + 6.45i)T + (16.8 - 95.5i)T^{2} \) |
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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
−11.33539931261790469641172805744, −9.820499188293096918981918850709, −9.253401386647542764129152899570, −8.227551533102123622288957304204, −7.00777359920308552339721715018, −5.81330529781203477058890232909, −4.92425209492888370987085756949, −4.02518223772307711421040785159, −2.54150974338285383709785553913, −0.77041225939115948095243038668,
2.55571431001233198482605641229, 3.49486296278533859498648661294, 4.91760053074950837347023512264, 5.85905378165193540878627368410, 6.80179192567040731075609352050, 7.69491448138426343931669858962, 8.519669475382176647862313775775, 9.955925452086510166594226866362, 10.75832129145301600632002893103, 11.57086163054770890267091071994
