| L(s) = 1 | + (−0.490 − 0.343i)2-s + (1.72 + 0.150i)3-s + (−0.561 − 1.54i)4-s + (−0.794 − 0.665i)6-s + (0.0636 + 0.136i)7-s + (−0.563 + 2.10i)8-s + (2.95 + 0.518i)9-s + (−3.54 − 4.23i)11-s + (−0.737 − 2.74i)12-s + (−1.08 − 1.54i)13-s + (0.0156 − 0.0887i)14-s + (−1.51 + 1.27i)16-s + (−1.18 − 4.40i)17-s + (−1.27 − 1.26i)18-s + (6.55 − 3.78i)19-s + ⋯ |
| L(s) = 1 | + (−0.346 − 0.242i)2-s + (0.996 + 0.0866i)3-s + (−0.280 − 0.771i)4-s + (−0.324 − 0.271i)6-s + (0.0240 + 0.0515i)7-s + (−0.199 + 0.744i)8-s + (0.984 + 0.172i)9-s + (−1.07 − 1.27i)11-s + (−0.212 − 0.792i)12-s + (−0.300 − 0.429i)13-s + (0.00418 − 0.0237i)14-s + (−0.379 + 0.318i)16-s + (−0.286 − 1.06i)17-s + (−0.299 − 0.298i)18-s + (1.50 − 0.867i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.230 + 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.230 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.885161 - 1.11878i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.885161 - 1.11878i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.72 - 0.150i)T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + (0.490 + 0.343i)T + (0.684 + 1.87i)T^{2} \) |
| 7 | \( 1 + (-0.0636 - 0.136i)T + (-4.49 + 5.36i)T^{2} \) |
| 11 | \( 1 + (3.54 + 4.23i)T + (-1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (1.08 + 1.54i)T + (-4.44 + 12.2i)T^{2} \) |
| 17 | \( 1 + (1.18 + 4.40i)T + (-14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (-6.55 + 3.78i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.883 - 0.412i)T + (14.7 + 17.6i)T^{2} \) |
| 29 | \( 1 + (0.486 + 2.75i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-1.70 + 0.622i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (5.71 - 1.53i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (0.682 + 0.120i)T + (38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (0.151 - 1.73i)T + (-42.3 - 7.46i)T^{2} \) |
| 47 | \( 1 + (10.0 - 4.70i)T + (30.2 - 36.0i)T^{2} \) |
| 53 | \( 1 + (0.204 + 0.204i)T + 53iT^{2} \) |
| 59 | \( 1 + (-0.796 - 0.668i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-9.12 - 3.32i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-8.96 + 6.27i)T + (22.9 - 62.9i)T^{2} \) |
| 71 | \( 1 + (2.53 + 1.46i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-9.95 - 2.66i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (11.9 - 2.10i)T + (74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (4.10 - 5.85i)T + (-28.3 - 77.9i)T^{2} \) |
| 89 | \( 1 + (-6.74 - 11.6i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (3.56 + 0.311i)T + (95.5 + 16.8i)T^{2} \) |
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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
−10.00304034977832812841456053717, −9.485789967916657118174454650688, −8.605300042296866862324606887509, −7.931410366484697697475694404022, −6.89434282993420651303663734341, −5.47023320054553718937136330435, −4.88203643442244975708062381079, −3.24769196843846900081626173246, −2.45548450977711912822693873613, −0.77267897371901376553990456622,
1.88740920747129897678880480409, 3.13235611301918025425986285773, 4.07673043099270537126935886435, 5.11463208808947909804048253792, 6.81197739515112441712874806882, 7.45782928698606171331310720636, 8.099485900534193749656025053015, 8.879368623097581787269249056752, 9.811485794953446422675969335170, 10.26505142487407071674985761022
