| L(s) = 1 | + (2.50 + 0.131i)2-s + (−0.971 − 1.20i)3-s + (4.26 + 0.448i)4-s + (−1.53 + 1.62i)5-s + (−2.27 − 3.13i)6-s + (2.61 − 0.377i)7-s + (5.68 + 0.899i)8-s + (0.127 − 0.601i)9-s + (−4.05 + 3.87i)10-s + (−5.16 + 1.09i)11-s + (−3.61 − 5.56i)12-s + (−2.31 − 1.17i)13-s + (6.60 − 0.602i)14-s + (3.44 + 0.255i)15-s + (5.71 + 1.21i)16-s + (0.412 + 1.07i)17-s + ⋯ |
| L(s) = 1 | + (1.77 + 0.0928i)2-s + (−0.561 − 0.692i)3-s + (2.13 + 0.224i)4-s + (−0.685 + 0.728i)5-s + (−0.929 − 1.27i)6-s + (0.989 − 0.142i)7-s + (2.00 + 0.318i)8-s + (0.0426 − 0.200i)9-s + (−1.28 + 1.22i)10-s + (−1.55 + 0.331i)11-s + (−1.04 − 1.60i)12-s + (−0.641 − 0.326i)13-s + (1.76 − 0.161i)14-s + (0.889 + 0.0660i)15-s + (1.42 + 0.303i)16-s + (0.100 + 0.260i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.193i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.981 + 0.193i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.30967 - 0.225414i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.30967 - 0.225414i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (1.53 - 1.62i)T \) |
| 7 | \( 1 + (-2.61 + 0.377i)T \) |
| good | 2 | \( 1 + (-2.50 - 0.131i)T + (1.98 + 0.209i)T^{2} \) |
| 3 | \( 1 + (0.971 + 1.20i)T + (-0.623 + 2.93i)T^{2} \) |
| 11 | \( 1 + (5.16 - 1.09i)T + (10.0 - 4.47i)T^{2} \) |
| 13 | \( 1 + (2.31 + 1.17i)T + (7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (-0.412 - 1.07i)T + (-12.6 + 11.3i)T^{2} \) |
| 19 | \( 1 + (-0.136 - 1.29i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (0.453 - 8.64i)T + (-22.8 - 2.40i)T^{2} \) |
| 29 | \( 1 + (-1.59 + 2.18i)T + (-8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-2.42 + 5.44i)T + (-20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (-2.70 + 1.75i)T + (15.0 - 33.8i)T^{2} \) |
| 41 | \( 1 + (-6.10 + 1.98i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (3.37 + 3.37i)T + 43iT^{2} \) |
| 47 | \( 1 + (-6.65 - 2.55i)T + (34.9 + 31.4i)T^{2} \) |
| 53 | \( 1 + (-2.65 + 2.14i)T + (11.0 - 51.8i)T^{2} \) |
| 59 | \( 1 + (-1.17 + 1.30i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + (-3.10 + 2.79i)T + (6.37 - 60.6i)T^{2} \) |
| 67 | \( 1 + (-0.225 + 0.0865i)T + (49.7 - 44.8i)T^{2} \) |
| 71 | \( 1 + (8.50 + 6.18i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (6.73 - 10.3i)T + (-29.6 - 66.6i)T^{2} \) |
| 79 | \( 1 + (-1.88 - 4.22i)T + (-52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (-1.46 + 9.26i)T + (-78.9 - 25.6i)T^{2} \) |
| 89 | \( 1 + (3.01 + 3.34i)T + (-9.30 + 88.5i)T^{2} \) |
| 97 | \( 1 + (-1.20 - 7.60i)T + (-92.2 + 29.9i)T^{2} \) |
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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
−12.72725636211556561142733255813, −11.84766348436997026926459963158, −11.33652308223909610889671618798, −10.29579687191245507324500651926, −7.61524687903706087305385458405, −7.48091960273327422528191302205, −6.02544092669229570872315258980, −5.14859146256801398995891472909, −3.88954559586875999386812792938, −2.41844300372131987112431682124,
2.63058344461571979132601863210, 4.49179630497031470603577863684, 4.79371808459749967302307987027, 5.65031349842384530072616844800, 7.34674739939828097504420818577, 8.436794413183827709883507595040, 10.40267274277463509447426060508, 11.08790335557612405714544427875, 11.93110545403336653686137749448, 12.68745073770488393843832508508
