| L(s) = 1 | + (0.841 + 0.540i)2-s + (−2.27 − 0.666i)3-s + (0.415 + 0.909i)4-s + (0.654 − 0.755i)5-s + (−1.55 − 1.78i)6-s + (1.31 + 0.845i)7-s + (−0.142 + 0.989i)8-s + (2.18 + 1.40i)9-s + (0.959 − 0.281i)10-s + (−0.427 + 0.492i)11-s + (−0.336 − 2.34i)12-s + (0.331 + 2.30i)13-s + (0.649 + 1.42i)14-s + (−1.99 + 1.27i)15-s + (−0.654 + 0.755i)16-s + (2.58 − 5.65i)17-s + ⋯ |
| L(s) = 1 | + (0.594 + 0.382i)2-s + (−1.31 − 0.385i)3-s + (0.207 + 0.454i)4-s + (0.292 − 0.337i)5-s + (−0.632 − 0.730i)6-s + (0.497 + 0.319i)7-s + (−0.0503 + 0.349i)8-s + (0.729 + 0.469i)9-s + (0.303 − 0.0890i)10-s + (−0.128 + 0.148i)11-s + (−0.0972 − 0.676i)12-s + (0.0920 + 0.639i)13-s + (0.173 + 0.380i)14-s + (−0.514 + 0.330i)15-s + (−0.163 + 0.188i)16-s + (0.625 − 1.37i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.877 - 0.480i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.877 - 0.480i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.47144 + 0.376257i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.47144 + 0.376257i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.841 - 0.540i)T \) |
| 5 | \( 1 + (-0.654 + 0.755i)T \) |
| 67 | \( 1 + (5.97 + 5.58i)T \) |
| good | 3 | \( 1 + (2.27 + 0.666i)T + (2.52 + 1.62i)T^{2} \) |
| 7 | \( 1 + (-1.31 - 0.845i)T + (2.90 + 6.36i)T^{2} \) |
| 11 | \( 1 + (0.427 - 0.492i)T + (-1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.331 - 2.30i)T + (-12.4 + 3.66i)T^{2} \) |
| 17 | \( 1 + (-2.58 + 5.65i)T + (-11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (1.63 - 1.04i)T + (7.89 - 17.2i)T^{2} \) |
| 23 | \( 1 + (-7.64 - 2.24i)T + (19.3 + 12.4i)T^{2} \) |
| 29 | \( 1 - 7.98T + 29T^{2} \) |
| 31 | \( 1 + (0.508 - 3.53i)T + (-29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 - 11.4T + 37T^{2} \) |
| 41 | \( 1 + (3.52 - 7.72i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (0.759 - 1.66i)T + (-28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 + (-6.10 - 1.79i)T + (39.5 + 25.4i)T^{2} \) |
| 53 | \( 1 + (-0.283 - 0.620i)T + (-34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (0.107 - 0.744i)T + (-56.6 - 16.6i)T^{2} \) |
| 61 | \( 1 + (9.12 + 10.5i)T + (-8.68 + 60.3i)T^{2} \) |
| 71 | \( 1 + (-2.68 - 5.86i)T + (-46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (2.66 + 3.08i)T + (-10.3 + 72.2i)T^{2} \) |
| 79 | \( 1 + (-0.308 - 2.14i)T + (-75.7 + 22.2i)T^{2} \) |
| 83 | \( 1 + (6.46 - 7.46i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (-2.31 + 0.680i)T + (74.8 - 48.1i)T^{2} \) |
| 97 | \( 1 - 15.4T + 97T^{2} \) |
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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
−10.98234076414126501054410426478, −9.764121538875881134627320819836, −8.777733114117813223493029194180, −7.65746789316048570802663807655, −6.76703573178772815533789222220, −6.05430880093007205849700735580, −5.04864457113628359987860757419, −4.71248321163768062082519939481, −2.88249940870469002660809539713, −1.20355640715303988041943053623,
1.01606033738072011632945768466, 2.76313572832199018110399940854, 4.12982824493320254981805054711, 4.97109799641969496181175259859, 5.83328764437648188834175015238, 6.44217139006607395623496013541, 7.64073923373022193878950831881, 8.833183739408557531399609289029, 10.27594600352800538552170887092, 10.51175967030192221230757609914
