| L(s) = 1 | + 0.848i·2-s + (1.44 − 2.62i)3-s + 3.27·4-s − 2.23i·5-s + (2.22 + 1.23i)6-s + 5.71·7-s + 6.17i·8-s + (−4.79 − 7.61i)9-s + 1.89·10-s + 0.158i·11-s + (4.75 − 8.61i)12-s − 3.60·13-s + 4.85i·14-s + (−5.87 − 3.24i)15-s + 7.87·16-s − 0.500i·17-s + ⋯ |
| L(s) = 1 | + 0.424i·2-s + (0.483 − 0.875i)3-s + 0.819·4-s − 0.447i·5-s + (0.371 + 0.205i)6-s + 0.816·7-s + 0.772i·8-s + (−0.532 − 0.846i)9-s + 0.189·10-s + 0.0143i·11-s + (0.396 − 0.717i)12-s − 0.277·13-s + 0.346i·14-s + (−0.391 − 0.216i)15-s + 0.492·16-s − 0.0294i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.875 + 0.483i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.875 + 0.483i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.13489 - 0.550200i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.13489 - 0.550200i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.44 + 2.62i)T \) |
| 5 | \( 1 + 2.23iT \) |
| 13 | \( 1 + 3.60T \) |
| good | 2 | \( 1 - 0.848iT - 4T^{2} \) |
| 7 | \( 1 - 5.71T + 49T^{2} \) |
| 11 | \( 1 - 0.158iT - 121T^{2} \) |
| 17 | \( 1 + 0.500iT - 289T^{2} \) |
| 19 | \( 1 + 7.38T + 361T^{2} \) |
| 23 | \( 1 + 27.1iT - 529T^{2} \) |
| 29 | \( 1 - 43.5iT - 841T^{2} \) |
| 31 | \( 1 - 60.8T + 961T^{2} \) |
| 37 | \( 1 + 16.3T + 1.36e3T^{2} \) |
| 41 | \( 1 - 38.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 33.5T + 1.84e3T^{2} \) |
| 47 | \( 1 - 10.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 35.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 58.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 49.4T + 3.72e3T^{2} \) |
| 67 | \( 1 + 0.886T + 4.48e3T^{2} \) |
| 71 | \( 1 - 70.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 91.7T + 5.32e3T^{2} \) |
| 79 | \( 1 + 99.0T + 6.24e3T^{2} \) |
| 83 | \( 1 + 7.11iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 38.6iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 143.T + 9.40e3T^{2} \) |
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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
−12.17634089382837172691632052814, −11.50299758960258856793822974047, −10.30723133275280055312645481070, −8.687089243886029165638835126245, −8.120338763508920782612890358461, −7.09008116380491430744286622091, −6.19150726172694594459568286209, −4.81708692432913245230989222556, −2.79879677817058305466985405247, −1.48200795594353538920033334688,
2.05056435038159901789549201568, 3.25454643518115818598827255975, 4.55370875878716002377703568285, 5.99598370346987659016462705081, 7.41664172063428962774150177938, 8.324865467413615502857751507243, 9.719805567227878054343510418384, 10.41316613416400637010867252250, 11.32580529524585472274207157060, 11.93036230020551373547313936552
