L(s) = 1 | − 0.0545·3-s − 5-s − 3.71·7-s − 2.99·9-s − 5.22·11-s + 0.0545·15-s − 2.97·17-s − 6.13·19-s + 0.202·21-s + 3.62·23-s + 25-s + 0.326·27-s − 2.03·29-s − 0.100·31-s + 0.284·33-s + 3.71·35-s − 7.32·37-s + 12.1·41-s + 8.18·43-s + 2.99·45-s + 8.41·47-s + 6.82·49-s + 0.162·51-s − 6.11·53-s + 5.22·55-s + 0.334·57-s − 1.53·59-s + ⋯ |
L(s) = 1 | − 0.0314·3-s − 0.447·5-s − 1.40·7-s − 0.999·9-s − 1.57·11-s + 0.0140·15-s − 0.722·17-s − 1.40·19-s + 0.0442·21-s + 0.755·23-s + 0.200·25-s + 0.0629·27-s − 0.377·29-s − 0.0180·31-s + 0.0495·33-s + 0.628·35-s − 1.20·37-s + 1.90·41-s + 1.24·43-s + 0.446·45-s + 1.22·47-s + 0.974·49-s + 0.0227·51-s − 0.839·53-s + 0.703·55-s + 0.0442·57-s − 0.199·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4014103884\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4014103884\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 3 | \( 1 + 0.0545T + 3T^{2} \) |
| 7 | \( 1 + 3.71T + 7T^{2} \) |
| 11 | \( 1 + 5.22T + 11T^{2} \) |
| 17 | \( 1 + 2.97T + 17T^{2} \) |
| 19 | \( 1 + 6.13T + 19T^{2} \) |
| 23 | \( 1 - 3.62T + 23T^{2} \) |
| 29 | \( 1 + 2.03T + 29T^{2} \) |
| 31 | \( 1 + 0.100T + 31T^{2} \) |
| 37 | \( 1 + 7.32T + 37T^{2} \) |
| 41 | \( 1 - 12.1T + 41T^{2} \) |
| 43 | \( 1 - 8.18T + 43T^{2} \) |
| 47 | \( 1 - 8.41T + 47T^{2} \) |
| 53 | \( 1 + 6.11T + 53T^{2} \) |
| 59 | \( 1 + 1.53T + 59T^{2} \) |
| 61 | \( 1 - 7.09T + 61T^{2} \) |
| 67 | \( 1 + 14.6T + 67T^{2} \) |
| 71 | \( 1 + 13.3T + 71T^{2} \) |
| 73 | \( 1 - 11.5T + 73T^{2} \) |
| 79 | \( 1 + 10.0T + 79T^{2} \) |
| 83 | \( 1 + 4.21T + 83T^{2} \) |
| 89 | \( 1 - 7.86T + 89T^{2} \) |
| 97 | \( 1 - 12.6T + 97T^{2} \) |
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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
−8.841534925906126554203752098184, −7.80797828782760316304193986653, −7.18993667324169546952189250555, −6.25446733322254425990385981123, −5.75513796011338057191163056170, −4.78571432405021799944716467002, −3.85594634330975439372655357650, −2.88564202127660553285656139232, −2.41866964867671312173733544877, −0.34428503770653198672297448576,
0.34428503770653198672297448576, 2.41866964867671312173733544877, 2.88564202127660553285656139232, 3.85594634330975439372655357650, 4.78571432405021799944716467002, 5.75513796011338057191163056170, 6.25446733322254425990385981123, 7.18993667324169546952189250555, 7.80797828782760316304193986653, 8.841534925906126554203752098184