| L(s) = 1 | + (1.72 − 0.117i)3-s − i·5-s + 0.723i·7-s + (2.97 − 0.404i)9-s + (−2.32 + 2.36i)11-s − 4.17i·13-s + (−0.117 − 1.72i)15-s + 4.49·17-s − 7.94i·19-s + (0.0846 + 1.24i)21-s + 8.91i·23-s − 25-s + (5.08 − 1.04i)27-s + 4.98·29-s + 5.85·31-s + ⋯ |
| L(s) = 1 | + (0.997 − 0.0675i)3-s − 0.447i·5-s + 0.273i·7-s + (0.990 − 0.134i)9-s + (−0.700 + 0.713i)11-s − 1.15i·13-s + (−0.0302 − 0.446i)15-s + 1.09·17-s − 1.82i·19-s + (0.0184 + 0.272i)21-s + 1.85i·23-s − 0.200·25-s + (0.979 − 0.201i)27-s + 0.926·29-s + 1.05·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.759 + 0.650i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.759 + 0.650i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.696657895\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.696657895\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.72 + 0.117i)T \) |
| 5 | \( 1 + iT \) |
| 11 | \( 1 + (2.32 - 2.36i)T \) |
| good | 7 | \( 1 - 0.723iT - 7T^{2} \) |
| 13 | \( 1 + 4.17iT - 13T^{2} \) |
| 17 | \( 1 - 4.49T + 17T^{2} \) |
| 19 | \( 1 + 7.94iT - 19T^{2} \) |
| 23 | \( 1 - 8.91iT - 23T^{2} \) |
| 29 | \( 1 - 4.98T + 29T^{2} \) |
| 31 | \( 1 - 5.85T + 31T^{2} \) |
| 37 | \( 1 + 5.54T + 37T^{2} \) |
| 41 | \( 1 + 3.53T + 41T^{2} \) |
| 43 | \( 1 + 10.4iT - 43T^{2} \) |
| 47 | \( 1 + 4.44iT - 47T^{2} \) |
| 53 | \( 1 + 2.41iT - 53T^{2} \) |
| 59 | \( 1 + 9.71iT - 59T^{2} \) |
| 61 | \( 1 - 2.59iT - 61T^{2} \) |
| 67 | \( 1 - 9.24T + 67T^{2} \) |
| 71 | \( 1 + 3.54iT - 71T^{2} \) |
| 73 | \( 1 - 4.56iT - 73T^{2} \) |
| 79 | \( 1 - 11.1iT - 79T^{2} \) |
| 83 | \( 1 - 15.3T + 83T^{2} \) |
| 89 | \( 1 + 2.10iT - 89T^{2} \) |
| 97 | \( 1 - 8.27T + 97T^{2} \) |
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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
−8.683676770472927484677477762842, −8.069631647035371667455246928676, −7.43629692220055406271665907560, −6.73697770299898792570335569233, −5.28301779048570148340901649940, −5.08373134301447737344451785855, −3.77534820230694658062752094593, −2.98958180338778428871583008760, −2.15785693020796979252558298968, −0.876213955070272641785123011344,
1.24244507319561031508824570752, 2.42406490268166475222079481014, 3.19573958135364787287280103492, 4.02100525915228649844404314540, 4.82844197642576659147236804473, 6.06863247611773438929886536183, 6.67505049103979213567491806036, 7.66681770080600133614253188276, 8.175394824472220032745275483389, 8.769303377619329766961355265039
