| L(s) = 1 | + (0.528 − 0.443i)2-s + (0.652 + 0.237i)3-s + (−0.264 + 1.50i)4-s + (0.173 + 0.984i)5-s + (0.450 − 0.164i)6-s + (1.16 − 2.02i)7-s + (1.21 + 2.10i)8-s + (−1.92 − 1.61i)9-s + (0.528 + 0.443i)10-s + (−2.28 − 3.96i)11-s + (−0.529 + 0.916i)12-s + (−1.20 + 0.438i)13-s + (−0.279 − 1.58i)14-s + (−0.120 + 0.684i)15-s + (−1.28 − 0.468i)16-s + (−0.501 + 0.420i)17-s + ⋯ |
| L(s) = 1 | + (0.373 − 0.313i)2-s + (0.376 + 0.137i)3-s + (−0.132 + 0.750i)4-s + (0.0776 + 0.440i)5-s + (0.183 − 0.0669i)6-s + (0.441 − 0.764i)7-s + (0.429 + 0.744i)8-s + (−0.642 − 0.539i)9-s + (0.167 + 0.140i)10-s + (−0.690 − 1.19i)11-s + (−0.152 + 0.264i)12-s + (−0.333 + 0.121i)13-s + (−0.0747 − 0.424i)14-s + (−0.0311 + 0.176i)15-s + (−0.321 − 0.117i)16-s + (−0.121 + 0.102i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.115i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.115i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.25192 + 0.0722573i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.25192 + 0.0722573i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-0.173 - 0.984i)T \) |
| 19 | \( 1 + (-3.67 - 2.34i)T \) |
| good | 2 | \( 1 + (-0.528 + 0.443i)T + (0.347 - 1.96i)T^{2} \) |
| 3 | \( 1 + (-0.652 - 0.237i)T + (2.29 + 1.92i)T^{2} \) |
| 7 | \( 1 + (-1.16 + 2.02i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (2.28 + 3.96i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1.20 - 0.438i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (0.501 - 0.420i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (0.966 - 5.48i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (3.62 + 3.04i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-2.24 + 3.88i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 7.79T + 37T^{2} \) |
| 41 | \( 1 + (8.17 + 2.97i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-1.66 - 9.44i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-4.84 - 4.06i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (-1.14 + 6.50i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (-4.51 + 3.78i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (1.30 - 7.38i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-10.0 - 8.39i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-0.651 - 3.69i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (7.48 + 2.72i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (5.92 + 2.15i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-4.91 + 8.51i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (11.4 - 4.16i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-3.22 + 2.70i)T + (16.8 - 95.5i)T^{2} \) |
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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
−13.85537494297990048038962065751, −13.24663224758600584015422540933, −11.70642711052697535938583288809, −11.15301680866560580634443944975, −9.707926158206654778125534595068, −8.299492392156813760997806031197, −7.52806396893408812164873273373, −5.70375312288071039060043918451, −3.96131752053887658939684312140, −2.89345669745828864413365376674,
2.26689143889831440333173002985, 4.80345278759043938481512574443, 5.48425107548733331343300915306, 7.14706907505468502393786420292, 8.446221767059002592537125169903, 9.546682720853121976406333954850, 10.68285867815505474257413162938, 12.07158768313527822998298351376, 13.13108354112366640442271820120, 14.07678871478744155261076836279
