| L(s) = 1 | + 4i·7-s − 2·11-s + 4i·13-s + i·17-s + 5·19-s − 5i·23-s + 8·29-s + 7·31-s + 6i·37-s − 6·41-s − 2i·43-s + 8i·47-s − 9·49-s − 9i·53-s + 4·59-s + ⋯ |
| L(s) = 1 | + 1.51i·7-s − 0.603·11-s + 1.10i·13-s + 0.242i·17-s + 1.14·19-s − 1.04i·23-s + 1.48·29-s + 1.25·31-s + 0.986i·37-s − 0.937·41-s − 0.304i·43-s + 1.16i·47-s − 1.28·49-s − 1.23i·53-s + 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.670908421\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.670908421\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 4iT - 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 - iT - 17T^{2} \) |
| 19 | \( 1 - 5T + 19T^{2} \) |
| 23 | \( 1 + 5iT - 23T^{2} \) |
| 29 | \( 1 - 8T + 29T^{2} \) |
| 31 | \( 1 - 7T + 31T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 2iT - 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 + 9iT - 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 - 13T + 61T^{2} \) |
| 67 | \( 1 - 10iT - 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 + 9T + 79T^{2} \) |
| 83 | \( 1 - 17iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 8iT - 97T^{2} \) |
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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
−8.457513604577591734207474731755, −7.912628774792344586356678604113, −6.64839939934742245942584932295, −6.48498415433447582404446566068, −5.34373350150100179052004926480, −4.98451315301008518332294637764, −4.00607476183968351693958705982, −2.80163996766485035779918020571, −2.44384031563773400378362663838, −1.21879466829391126529469352461,
0.49394191554712047749289178925, 1.28163865748179823440418228252, 2.73570912604652435023129315319, 3.38695235142391045594797925299, 4.22034593240676834943524441166, 5.07645861724610998857834536149, 5.63898729066692991743344834641, 6.71930137280208014404550719121, 7.27377081650651491099368540753, 7.888779604446513700235520831823
