| L(s) = 1 | + (−0.651 − 0.816i)2-s + (0.922 + 2.34i)3-s + (0.202 − 0.885i)4-s + (−0.0373 + 0.498i)5-s + (1.31 − 2.28i)6-s + (−1.65 − 2.86i)7-s + (−2.73 + 1.31i)8-s + (−2.47 + 2.29i)9-s + (0.431 − 0.294i)10-s + (0.828 + 3.62i)11-s + (2.26 − 0.341i)12-s + (−3.87 − 2.64i)13-s + (−1.26 + 3.21i)14-s + (−1.20 + 0.371i)15-s + (1.22 + 0.589i)16-s + (−0.343 − 4.57i)17-s + ⋯ |
| L(s) = 1 | + (−0.460 − 0.577i)2-s + (0.532 + 1.35i)3-s + (0.101 − 0.442i)4-s + (−0.0167 + 0.222i)5-s + (0.538 − 0.932i)6-s + (−0.624 − 1.08i)7-s + (−0.967 + 0.466i)8-s + (−0.823 + 0.764i)9-s + (0.136 − 0.0930i)10-s + (0.249 + 1.09i)11-s + (0.654 − 0.0986i)12-s + (−1.07 − 0.732i)13-s + (−0.337 + 0.859i)14-s + (−0.311 + 0.0960i)15-s + (0.305 + 0.147i)16-s + (−0.0831 − 1.11i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0712i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.709139 - 0.0252997i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.709139 - 0.0252997i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-6.26 - 1.92i)T \) |
| good | 2 | \( 1 + (0.651 + 0.816i)T + (-0.445 + 1.94i)T^{2} \) |
| 3 | \( 1 + (-0.922 - 2.34i)T + (-2.19 + 2.04i)T^{2} \) |
| 5 | \( 1 + (0.0373 - 0.498i)T + (-4.94 - 0.745i)T^{2} \) |
| 7 | \( 1 + (1.65 + 2.86i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.828 - 3.62i)T + (-9.91 + 4.77i)T^{2} \) |
| 13 | \( 1 + (3.87 + 2.64i)T + (4.74 + 12.1i)T^{2} \) |
| 17 | \( 1 + (0.343 + 4.57i)T + (-16.8 + 2.53i)T^{2} \) |
| 19 | \( 1 + (-4.44 - 4.12i)T + (1.41 + 18.9i)T^{2} \) |
| 23 | \( 1 + (0.371 + 0.114i)T + (19.0 + 12.9i)T^{2} \) |
| 29 | \( 1 + (1.29 - 3.29i)T + (-21.2 - 19.7i)T^{2} \) |
| 31 | \( 1 + (-0.861 + 0.129i)T + (29.6 - 9.13i)T^{2} \) |
| 37 | \( 1 + (1.96 - 3.40i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (6.72 + 8.43i)T + (-9.12 + 39.9i)T^{2} \) |
| 47 | \( 1 + (0.776 - 3.40i)T + (-42.3 - 20.3i)T^{2} \) |
| 53 | \( 1 + (-3.94 + 2.68i)T + (19.3 - 49.3i)T^{2} \) |
| 59 | \( 1 + (6.05 + 2.91i)T + (36.7 + 46.1i)T^{2} \) |
| 61 | \( 1 + (5.18 + 0.782i)T + (58.2 + 17.9i)T^{2} \) |
| 67 | \( 1 + (7.73 + 7.17i)T + (5.00 + 66.8i)T^{2} \) |
| 71 | \( 1 + (-0.859 + 0.265i)T + (58.6 - 39.9i)T^{2} \) |
| 73 | \( 1 + (0.798 + 0.544i)T + (26.6 + 67.9i)T^{2} \) |
| 79 | \( 1 + (-0.500 - 0.867i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (1.24 + 3.17i)T + (-60.8 + 56.4i)T^{2} \) |
| 89 | \( 1 + (0.672 + 1.71i)T + (-65.2 + 60.5i)T^{2} \) |
| 97 | \( 1 + (0.213 + 0.936i)T + (-87.3 + 42.0i)T^{2} \) |
| 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
−15.86445550026294444604884465290, −14.88972951595294269873900116371, −14.08205403372406528374435551423, −12.20045451861484728016500051899, −10.58860523752104873676597569150, −9.991557700351727751689529686289, −9.309328748172286981697204593091, −7.22974631711017457486682902709, −4.94582305626491947305856745732, −3.17966145664464763119425229737,
2.81110447078383814848124626173, 6.14276547953100547990607515674, 7.20792796097456308017256313059, 8.454509187819405584679614690320, 9.210205040509821794413450110109, 11.80559141485463960891385417015, 12.53693245791985938283728951860, 13.56309044234343189139956223878, 14.95872205216282782626460012821, 16.18299620135581693796233233453
