L(s) = 1 | + (−0.623 − 0.781i)2-s + (1.14 − 1.43i)3-s + (−0.222 + 0.974i)4-s + (−0.900 + 0.433i)5-s − 1.82·6-s + 4.80·7-s + (0.900 − 0.433i)8-s + (−0.0773 − 0.338i)9-s + (0.900 + 0.433i)10-s + (1.16 + 5.10i)11-s + (1.14 + 1.43i)12-s + (3.55 − 1.71i)13-s + (−2.99 − 3.75i)14-s + (−0.407 + 1.78i)15-s + (−0.900 − 0.433i)16-s + (−3.09 − 1.49i)17-s + ⋯ |
L(s) = 1 | + (−0.440 − 0.552i)2-s + (0.658 − 0.825i)3-s + (−0.111 + 0.487i)4-s + (−0.402 + 0.194i)5-s − 0.746·6-s + 1.81·7-s + (0.318 − 0.153i)8-s + (−0.0257 − 0.112i)9-s + (0.284 + 0.137i)10-s + (0.351 + 1.53i)11-s + (0.329 + 0.412i)12-s + (0.984 − 0.474i)13-s + (−0.800 − 1.00i)14-s + (−0.105 + 0.460i)15-s + (−0.225 − 0.108i)16-s + (−0.751 − 0.361i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.603 + 0.797i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.603 + 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.39682 - 0.695069i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.39682 - 0.695069i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.623 + 0.781i)T \) |
| 5 | \( 1 + (0.900 - 0.433i)T \) |
| 43 | \( 1 + (2.24 + 6.15i)T \) |
good | 3 | \( 1 + (-1.14 + 1.43i)T + (-0.667 - 2.92i)T^{2} \) |
| 7 | \( 1 - 4.80T + 7T^{2} \) |
| 11 | \( 1 + (-1.16 - 5.10i)T + (-9.91 + 4.77i)T^{2} \) |
| 13 | \( 1 + (-3.55 + 1.71i)T + (8.10 - 10.1i)T^{2} \) |
| 17 | \( 1 + (3.09 + 1.49i)T + (10.5 + 13.2i)T^{2} \) |
| 19 | \( 1 + (0.793 - 3.47i)T + (-17.1 - 8.24i)T^{2} \) |
| 23 | \( 1 + (1.48 + 6.51i)T + (-20.7 + 9.97i)T^{2} \) |
| 29 | \( 1 + (-0.375 - 0.471i)T + (-6.45 + 28.2i)T^{2} \) |
| 31 | \( 1 + (2.24 + 2.81i)T + (-6.89 + 30.2i)T^{2} \) |
| 37 | \( 1 + 7.64T + 37T^{2} \) |
| 41 | \( 1 + (5.28 + 6.62i)T + (-9.12 + 39.9i)T^{2} \) |
| 47 | \( 1 + (2.39 - 10.4i)T + (-42.3 - 20.3i)T^{2} \) |
| 53 | \( 1 + (-5.00 - 2.40i)T + (33.0 + 41.4i)T^{2} \) |
| 59 | \( 1 + (7.64 + 3.68i)T + (36.7 + 46.1i)T^{2} \) |
| 61 | \( 1 + (-7.07 + 8.87i)T + (-13.5 - 59.4i)T^{2} \) |
| 67 | \( 1 + (1.06 - 4.64i)T + (-60.3 - 29.0i)T^{2} \) |
| 71 | \( 1 + (-0.106 + 0.467i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (5.59 - 2.69i)T + (45.5 - 57.0i)T^{2} \) |
| 79 | \( 1 - 0.197T + 79T^{2} \) |
| 83 | \( 1 + (-8.50 + 10.6i)T + (-18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (7.76 - 9.73i)T + (-19.8 - 86.7i)T^{2} \) |
| 97 | \( 1 + (-1.36 - 5.96i)T + (-87.3 + 42.0i)T^{2} \) |
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
−10.94907886644951111696180894316, −10.38619357137974935204963231406, −8.895217500962253467711446649380, −8.262960965670631193492553431173, −7.63759535223161569241072205254, −6.79737985673530218767590993669, −4.95755697335993728221675460564, −3.99076820374091713751415341699, −2.26660974662621342625808691374, −1.56682182178584585265538012384,
1.46184988280325867241812526689, 3.51466633577397085825686453347, 4.44678589347408812215095371233, 5.47042660002620741850623610569, 6.73593495081125828294577305496, 8.048465478559283369252749416957, 8.681870321759523081548266444184, 8.954216095117243493357326415519, 10.38295247609252804832211868097, 11.29276778118475032140291428533