| 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
−11.57086163054770890267091071994, −10.75832129145301600632002893103, −9.955925452086510166594226866362, −8.519669475382176647862313775775, −7.69491448138426343931669858962, −6.80179192567040731075609352050, −5.85905378165193540878627368410, −4.91760053074950837347023512264, −3.49486296278533859498648661294, −2.55571431001233198482605641229,
0.77041225939115948095243038668, 2.54150974338285383709785553913, 4.02518223772307711421040785159, 4.92425209492888370987085756949, 5.81330529781203477058890232909, 7.00777359920308552339721715018, 8.227551533102123622288957304204, 9.253401386647542764129152899570, 9.820499188293096918981918850709, 11.33539931261790469641172805744
