| L(s) = 1 | + i·3-s + (−1.48 − 1.67i)5-s − 1.19i·7-s − 9-s − 11-s − 0.806i·13-s + (1.67 − 1.48i)15-s + 3.76i·17-s − 5.35·19-s + 1.19·21-s + 4i·23-s + (−0.612 + 4.96i)25-s − i·27-s + 4.31·29-s − 0.962·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + (−0.662 − 0.749i)5-s − 0.451i·7-s − 0.333·9-s − 0.301·11-s − 0.223i·13-s + (0.432 − 0.382i)15-s + 0.913i·17-s − 1.22·19-s + 0.260·21-s + 0.834i·23-s + (−0.122 + 0.992i)25-s − 0.192i·27-s + 0.800·29-s − 0.172·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 - 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.749 - 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.234408995\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.234408995\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (1.48 + 1.67i)T \) |
| 11 | \( 1 + T \) |
| good | 7 | \( 1 + 1.19iT - 7T^{2} \) |
| 13 | \( 1 + 0.806iT - 13T^{2} \) |
| 17 | \( 1 - 3.76iT - 17T^{2} \) |
| 19 | \( 1 + 5.35T + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 - 4.31T + 29T^{2} \) |
| 31 | \( 1 + 0.962T + 31T^{2} \) |
| 37 | \( 1 - 1.61iT - 37T^{2} \) |
| 41 | \( 1 - 9.08T + 41T^{2} \) |
| 43 | \( 1 - 4.41iT - 43T^{2} \) |
| 47 | \( 1 + 12.3iT - 47T^{2} \) |
| 53 | \( 1 + 1.42iT - 53T^{2} \) |
| 59 | \( 1 - 13.2T + 59T^{2} \) |
| 61 | \( 1 + 0.0752T + 61T^{2} \) |
| 67 | \( 1 - 2.70iT - 67T^{2} \) |
| 71 | \( 1 - 14.0T + 71T^{2} \) |
| 73 | \( 1 - 10.7iT - 73T^{2} \) |
| 79 | \( 1 - 13.9T + 79T^{2} \) |
| 83 | \( 1 + 9.89iT - 83T^{2} \) |
| 89 | \( 1 + 16.8T + 89T^{2} \) |
| 97 | \( 1 - 11.4iT - 97T^{2} \) |
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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
−8.772725883677591832124471082981, −8.329018101570817948272871827236, −7.61036721483342525558845958042, −6.66890331415702974650305628050, −5.69306686036635602482810992159, −4.94105589999368435064477576306, −4.08364994614778861624053789734, −3.62362379909875747507960830670, −2.26356132652602173684245305026, −0.844845720571783865139902330449,
0.55298553805198929717913163165, 2.27021186737293786338344045180, 2.77368565703586902391053758035, 3.95386488446155801072114187920, 4.78768635491278309561050536293, 5.89284410021160953758037319343, 6.58401394103575596046598262579, 7.21657901981280274939950388017, 7.980851406146128205305365973755, 8.608759176960002325258124045259
