| L(s) = 1 | + (0.173 + 0.984i)2-s + (−0.173 − 0.984i)3-s + (−0.939 + 0.342i)4-s + (3.08 + 1.12i)5-s + (0.939 − 0.342i)6-s + (0.0876 + 2.64i)7-s + (−0.5 − 0.866i)8-s + (−0.939 + 0.342i)9-s + (−0.569 + 3.23i)10-s + (−2.54 + 4.40i)11-s + (0.5 + 0.866i)12-s + (−5.73 + 2.08i)13-s + (−2.58 + 0.545i)14-s + (0.569 − 3.23i)15-s + (0.766 − 0.642i)16-s + (−3.93 − 1.43i)17-s + ⋯ |
| L(s) = 1 | + (0.122 + 0.696i)2-s + (−0.100 − 0.568i)3-s + (−0.469 + 0.171i)4-s + (1.37 + 0.501i)5-s + (0.383 − 0.139i)6-s + (0.0331 + 0.999i)7-s + (−0.176 − 0.306i)8-s + (−0.313 + 0.114i)9-s + (−0.180 + 1.02i)10-s + (−0.766 + 1.32i)11-s + (0.144 + 0.249i)12-s + (−1.58 + 0.578i)13-s + (−0.691 + 0.145i)14-s + (0.147 − 0.834i)15-s + (0.191 − 0.160i)16-s + (−0.953 − 0.347i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.651 - 0.758i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.651 - 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.599455 + 1.30566i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.599455 + 1.30566i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.173 - 0.984i)T \) |
| 3 | \( 1 + (0.173 + 0.984i)T \) |
| 7 | \( 1 + (-0.0876 - 2.64i)T \) |
| 19 | \( 1 + (-4.33 + 0.456i)T \) |
| good | 5 | \( 1 + (-3.08 - 1.12i)T + (3.83 + 3.21i)T^{2} \) |
| 11 | \( 1 + (2.54 - 4.40i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (5.73 - 2.08i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (3.93 + 1.43i)T + (13.0 + 10.9i)T^{2} \) |
| 23 | \( 1 + (0.993 + 0.834i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (1.41 + 1.18i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 - 7.45T + 31T^{2} \) |
| 37 | \( 1 + (4.32 - 7.49i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-6.80 - 2.47i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (1.36 + 7.74i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (1.21 - 0.442i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (-3.11 + 1.13i)T + (40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (-10.2 - 3.73i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (4.25 + 3.57i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-0.730 + 4.14i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-2.55 - 14.4i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-0.731 - 4.14i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (-13.2 + 11.0i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-2.17 + 3.77i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (2.45 - 13.9i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (2.44 - 2.05i)T + (16.8 - 95.5i)T^{2} \) |
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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
−10.15923172555944746560536941111, −9.728029013595154824522243130440, −8.950866961489700900656600932206, −7.75426773776537077614751360428, −6.96116558260141552225571058940, −6.36499322026188070334633151433, −5.27921664874258121251111968893, −4.82250377808414062718572799011, −2.58188503693020393150483528499, −2.15195529970947822138019613632,
0.65939383522014522588522081450, 2.26753542911102236174052845799, 3.30507554965180179328648768800, 4.62233026232336623522855671167, 5.31394429895081068220844234921, 6.08692973683252431747054622268, 7.49102874521378148741397317474, 8.532044615904597783968128798202, 9.488837281409763366740426531642, 10.02412730610915984840601550841
