| L(s) = 1 | + 0.618·3-s − 1.61·7-s − 2.61·9-s − 5.52·11-s − 5.52·13-s + 3.41·17-s + 3.41·19-s − 1.00·21-s + 2.38·23-s − 3.47·27-s + 1.09·29-s + 5.52·31-s − 3.41·33-s − 8.93·37-s − 3.41·39-s + 5.85·41-s + 12.3·43-s − 1.09·47-s − 4.38·49-s + 2.10·51-s + 11.0·53-s + 2.10·57-s − 12.3·59-s + 1.14·61-s + 4.23·63-s + 10.4·67-s + 1.47·69-s + ⋯ |
| L(s) = 1 | + 0.356·3-s − 0.611·7-s − 0.872·9-s − 1.66·11-s − 1.53·13-s + 0.827·17-s + 0.782·19-s − 0.218·21-s + 0.496·23-s − 0.668·27-s + 0.202·29-s + 0.991·31-s − 0.593·33-s − 1.46·37-s − 0.546·39-s + 0.914·41-s + 1.87·43-s − 0.159·47-s − 0.625·49-s + 0.295·51-s + 1.51·53-s + 0.279·57-s − 1.60·59-s + 0.146·61-s + 0.533·63-s + 1.27·67-s + 0.177·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.209607492\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.209607492\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 0.618T + 3T^{2} \) |
| 7 | \( 1 + 1.61T + 7T^{2} \) |
| 11 | \( 1 + 5.52T + 11T^{2} \) |
| 13 | \( 1 + 5.52T + 13T^{2} \) |
| 17 | \( 1 - 3.41T + 17T^{2} \) |
| 19 | \( 1 - 3.41T + 19T^{2} \) |
| 23 | \( 1 - 2.38T + 23T^{2} \) |
| 29 | \( 1 - 1.09T + 29T^{2} \) |
| 31 | \( 1 - 5.52T + 31T^{2} \) |
| 37 | \( 1 + 8.93T + 37T^{2} \) |
| 41 | \( 1 - 5.85T + 41T^{2} \) |
| 43 | \( 1 - 12.3T + 43T^{2} \) |
| 47 | \( 1 + 1.09T + 47T^{2} \) |
| 53 | \( 1 - 11.0T + 53T^{2} \) |
| 59 | \( 1 + 12.3T + 59T^{2} \) |
| 61 | \( 1 - 1.14T + 61T^{2} \) |
| 67 | \( 1 - 10.4T + 67T^{2} \) |
| 71 | \( 1 - 3.41T + 71T^{2} \) |
| 73 | \( 1 - 12.3T + 73T^{2} \) |
| 79 | \( 1 - 8.93T + 79T^{2} \) |
| 83 | \( 1 - 12.5T + 83T^{2} \) |
| 89 | \( 1 - 3.14T + 89T^{2} \) |
| 97 | \( 1 + 12.3T + 97T^{2} \) |
| 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
−8.307957796034639869688040647966, −7.72086625110724206428357176779, −7.22630048649340134548167799237, −6.17332904302215755608763510102, −5.28881237529974181631551763333, −4.98473523052769131967442592354, −3.60497298080258198980052572396, −2.78247817031101696442145548398, −2.40714561161291904105615398088, −0.58022852315035591905303209686,
0.58022852315035591905303209686, 2.40714561161291904105615398088, 2.78247817031101696442145548398, 3.60497298080258198980052572396, 4.98473523052769131967442592354, 5.28881237529974181631551763333, 6.17332904302215755608763510102, 7.22630048649340134548167799237, 7.72086625110724206428357176779, 8.307957796034639869688040647966
