| L(s) = 1 | + 2-s + 4-s + 4.34·5-s − 7-s + 8-s + 4.34·10-s + 3.07·11-s + 6.34·13-s − 14-s + 16-s − 1.07·17-s − 19-s + 4.34·20-s + 3.07·22-s − 2.34·23-s + 13.8·25-s + 6.34·26-s − 28-s − 8.83·29-s − 5.41·31-s + 32-s − 1.07·34-s − 4.34·35-s − 3.41·37-s − 38-s + 4.34·40-s − 6.68·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.94·5-s − 0.377·7-s + 0.353·8-s + 1.37·10-s + 0.928·11-s + 1.75·13-s − 0.267·14-s + 0.250·16-s − 0.261·17-s − 0.229·19-s + 0.970·20-s + 0.656·22-s − 0.487·23-s + 2.76·25-s + 1.24·26-s − 0.188·28-s − 1.64·29-s − 0.973·31-s + 0.176·32-s − 0.184·34-s − 0.733·35-s − 0.562·37-s − 0.162·38-s + 0.686·40-s − 1.04·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.320999253\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.320999253\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 - 4.34T + 5T^{2} \) |
| 11 | \( 1 - 3.07T + 11T^{2} \) |
| 13 | \( 1 - 6.34T + 13T^{2} \) |
| 17 | \( 1 + 1.07T + 17T^{2} \) |
| 23 | \( 1 + 2.34T + 23T^{2} \) |
| 29 | \( 1 + 8.83T + 29T^{2} \) |
| 31 | \( 1 + 5.41T + 31T^{2} \) |
| 37 | \( 1 + 3.41T + 37T^{2} \) |
| 41 | \( 1 + 6.68T + 41T^{2} \) |
| 43 | \( 1 + 10.8T + 43T^{2} \) |
| 47 | \( 1 - 2.73T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 12.8T + 61T^{2} \) |
| 67 | \( 1 - 1.07T + 67T^{2} \) |
| 71 | \( 1 - 10.1T + 71T^{2} \) |
| 73 | \( 1 + 4.15T + 73T^{2} \) |
| 79 | \( 1 + 14.8T + 79T^{2} \) |
| 83 | \( 1 + 2.68T + 83T^{2} \) |
| 89 | \( 1 - 3.84T + 89T^{2} \) |
| 97 | \( 1 - 5.60T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.046541752620467877296601392492, −8.425795573557592041438899804848, −6.89055091181076924624750752190, −6.53914857205557580946783640968, −5.75270634618163321800799109803, −5.35120023269089736971651084965, −4.01236380573935747466359478896, −3.31777824065216942327734139276, −2.02999430096934033186869330473, −1.45563058974872740056334224117,
1.45563058974872740056334224117, 2.02999430096934033186869330473, 3.31777824065216942327734139276, 4.01236380573935747466359478896, 5.35120023269089736971651084965, 5.75270634618163321800799109803, 6.53914857205557580946783640968, 6.89055091181076924624750752190, 8.425795573557592041438899804848, 9.046541752620467877296601392492
