| L(s) = 1 | + (−0.132 + 0.407i)2-s + (1.24 + 0.265i)3-s + (1.46 + 1.06i)4-s + (0.5 + 0.866i)5-s + (−0.273 + 0.473i)6-s + (−2.12 + 0.944i)7-s + (−1.32 + 0.960i)8-s + (−1.25 − 0.557i)9-s + (−0.419 + 0.0890i)10-s + (0.574 − 5.46i)11-s + (1.55 + 1.72i)12-s + (1.15 − 1.28i)13-s + (−0.104 − 0.989i)14-s + (0.394 + 1.21i)15-s + (0.906 + 2.78i)16-s + (0.248 + 2.36i)17-s + ⋯ |
| L(s) = 1 | + (−0.0936 + 0.288i)2-s + (0.720 + 0.153i)3-s + (0.734 + 0.533i)4-s + (0.223 + 0.387i)5-s + (−0.111 + 0.193i)6-s + (−0.802 + 0.357i)7-s + (−0.467 + 0.339i)8-s + (−0.417 − 0.185i)9-s + (−0.132 + 0.0281i)10-s + (0.173 − 1.64i)11-s + (0.447 + 0.497i)12-s + (0.321 − 0.356i)13-s + (−0.0277 − 0.264i)14-s + (0.101 + 0.313i)15-s + (0.226 + 0.697i)16-s + (0.0602 + 0.573i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.622 - 0.782i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.622 - 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.27461 + 0.614725i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.27461 + 0.614725i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 31 | \( 1 + (-5.50 - 0.811i)T \) |
| good | 2 | \( 1 + (0.132 - 0.407i)T + (-1.61 - 1.17i)T^{2} \) |
| 3 | \( 1 + (-1.24 - 0.265i)T + (2.74 + 1.22i)T^{2} \) |
| 7 | \( 1 + (2.12 - 0.944i)T + (4.68 - 5.20i)T^{2} \) |
| 11 | \( 1 + (-0.574 + 5.46i)T + (-10.7 - 2.28i)T^{2} \) |
| 13 | \( 1 + (-1.15 + 1.28i)T + (-1.35 - 12.9i)T^{2} \) |
| 17 | \( 1 + (-0.248 - 2.36i)T + (-16.6 + 3.53i)T^{2} \) |
| 19 | \( 1 + (-0.0615 - 0.0683i)T + (-1.98 + 18.8i)T^{2} \) |
| 23 | \( 1 + (-2.64 + 1.92i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-2.98 + 9.17i)T + (-23.4 - 17.0i)T^{2} \) |
| 37 | \( 1 + (5.51 - 9.54i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (5.14 - 1.09i)T + (37.4 - 16.6i)T^{2} \) |
| 43 | \( 1 + (3.49 + 3.88i)T + (-4.49 + 42.7i)T^{2} \) |
| 47 | \( 1 + (-0.795 - 2.44i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (3.55 + 1.58i)T + (35.4 + 39.3i)T^{2} \) |
| 59 | \( 1 + (6.13 + 1.30i)T + (53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 + 0.530T + 61T^{2} \) |
| 67 | \( 1 + (-4.91 - 8.50i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.689 - 0.306i)T + (47.5 + 52.7i)T^{2} \) |
| 73 | \( 1 + (0.505 - 4.80i)T + (-71.4 - 15.1i)T^{2} \) |
| 79 | \( 1 + (-0.0774 - 0.737i)T + (-77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (5.69 - 1.20i)T + (75.8 - 33.7i)T^{2} \) |
| 89 | \( 1 + (-13.8 - 10.0i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (12.7 + 9.23i)T + (29.9 + 92.2i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.29232483268282388334957741785, −12.01385038355205357204516505688, −11.16651151703657996558015378415, −9.979976681610086422712535190886, −8.614878374543599800575901907319, −8.208728797257755842374526154846, −6.52369520152665799053972061023, −5.96496064032838217175103745830, −3.42704516925171875965907137637, −2.81193819027303444900677405814,
1.84256320730188505571853292105, 3.21657813993786164675320745332, 5.05809859586940929031062395367, 6.59009737667803932733891927059, 7.39876115998899946854590815559, 8.977819944848353201778272699840, 9.704734292851705829844544726027, 10.65849883658477392606923348404, 11.90535532098010773271485285869, 12.75731116175310947841475314487
