| L(s) = 1 | + (−1.98 − 4.58i)5-s + 0.667i·7-s − 19.4i·11-s − 20.0i·13-s + 16.3·17-s + 2.43·19-s + 34.3·23-s + (−17.1 + 18.2i)25-s − 5.73i·29-s − 26.8·31-s + (3.06 − 1.32i)35-s − 11.3i·37-s + 29.2i·41-s + 34.1i·43-s − 67.9·47-s + ⋯ |
| L(s) = 1 | + (−0.397 − 0.917i)5-s + 0.0954i·7-s − 1.76i·11-s − 1.54i·13-s + 0.960·17-s + 0.128·19-s + 1.49·23-s + (−0.684 + 0.728i)25-s − 0.197i·29-s − 0.866·31-s + (0.0875 − 0.0378i)35-s − 0.307i·37-s + 0.713i·41-s + 0.793i·43-s − 1.44·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.917 + 0.397i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.917 + 0.397i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.404000840\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.404000840\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.98 + 4.58i)T \) |
| good | 7 | \( 1 - 0.667iT - 49T^{2} \) |
| 11 | \( 1 + 19.4iT - 121T^{2} \) |
| 13 | \( 1 + 20.0iT - 169T^{2} \) |
| 17 | \( 1 - 16.3T + 289T^{2} \) |
| 19 | \( 1 - 2.43T + 361T^{2} \) |
| 23 | \( 1 - 34.3T + 529T^{2} \) |
| 29 | \( 1 + 5.73iT - 841T^{2} \) |
| 31 | \( 1 + 26.8T + 961T^{2} \) |
| 37 | \( 1 + 11.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 29.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 34.1iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 67.9T + 2.20e3T^{2} \) |
| 53 | \( 1 - 37.0T + 2.80e3T^{2} \) |
| 59 | \( 1 + 19.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 113.T + 3.72e3T^{2} \) |
| 67 | \( 1 + 106. iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 37.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 115. iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 16.7T + 6.24e3T^{2} \) |
| 83 | \( 1 - 90.4T + 6.88e3T^{2} \) |
| 89 | \( 1 + 28.0iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 9.42iT - 9.40e3T^{2} \) |
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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
−8.818212259290652102421348487810, −8.035797980784912644345837127966, −7.61896803921573738499620398229, −6.21311146358042972894391219350, −5.50190210777616960286570109949, −4.90804591908387078342233678614, −3.50978026392420785826967954957, −3.05386856213471921719340858227, −1.20225396231761378083691277949, −0.42093773288122961420571988832,
1.54657862690408781936965317703, 2.55072451452822135261040597375, 3.68744271270958947602879660893, 4.46095319571871962691807201385, 5.39779218318790901778651679604, 6.71646143327258964932419248311, 7.07110912948336785483483426168, 7.68337535627251215705948364693, 8.906228702716079376978416795633, 9.579473655826298351932802023226
