L(s) = 1 | + 1.19·2-s + 3-s − 0.561·4-s + 0.572·5-s + 1.19·6-s − 3.07·8-s + 9-s + 0.686·10-s + 2.52·11-s − 0.561·12-s + 5.05·13-s + 0.572·15-s − 2.56·16-s + 2.25·17-s + 1.19·18-s − 2.39·19-s − 0.321·20-s + 3.02·22-s + 23-s − 3.07·24-s − 4.67·25-s + 6.05·26-s + 27-s + 0.796·29-s + 0.686·30-s + 3.45·31-s + 3.07·32-s + ⋯ |
L(s) = 1 | + 0.847·2-s + 0.577·3-s − 0.280·4-s + 0.256·5-s + 0.489·6-s − 1.08·8-s + 0.333·9-s + 0.217·10-s + 0.761·11-s − 0.162·12-s + 1.40·13-s + 0.147·15-s − 0.640·16-s + 0.546·17-s + 0.282·18-s − 0.550·19-s − 0.0719·20-s + 0.645·22-s + 0.208·23-s − 0.627·24-s − 0.934·25-s + 1.18·26-s + 0.192·27-s + 0.147·29-s + 0.125·30-s + 0.620·31-s + 0.543·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.548354271\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.548354271\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 2 | \( 1 - 1.19T + 2T^{2} \) |
| 5 | \( 1 - 0.572T + 5T^{2} \) |
| 11 | \( 1 - 2.52T + 11T^{2} \) |
| 13 | \( 1 - 5.05T + 13T^{2} \) |
| 17 | \( 1 - 2.25T + 17T^{2} \) |
| 19 | \( 1 + 2.39T + 19T^{2} \) |
| 29 | \( 1 - 0.796T + 29T^{2} \) |
| 31 | \( 1 - 3.45T + 31T^{2} \) |
| 37 | \( 1 - 8.80T + 37T^{2} \) |
| 41 | \( 1 + 10.9T + 41T^{2} \) |
| 43 | \( 1 + 1.90T + 43T^{2} \) |
| 47 | \( 1 - 1.93T + 47T^{2} \) |
| 53 | \( 1 - 9.57T + 53T^{2} \) |
| 59 | \( 1 - 4.40T + 59T^{2} \) |
| 61 | \( 1 + 2.12T + 61T^{2} \) |
| 67 | \( 1 - 4.01T + 67T^{2} \) |
| 71 | \( 1 - 0.424T + 71T^{2} \) |
| 73 | \( 1 - 14.0T + 73T^{2} \) |
| 79 | \( 1 - 4.97T + 79T^{2} \) |
| 83 | \( 1 - 10.1T + 83T^{2} \) |
| 89 | \( 1 + 3.82T + 89T^{2} \) |
| 97 | \( 1 + 15.4T + 97T^{2} \) |
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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
−8.484050776951456418762019193957, −8.169229644516153037670053751082, −6.88278312250640338906417065967, −6.22575304164144736080388268033, −5.59922924298147321401904760123, −4.61360105668849927513139254339, −3.84145774980151186199174462061, −3.38334279239515921931826446830, −2.24394479268618871659796885667, −1.01957132746078541012230013810,
1.01957132746078541012230013810, 2.24394479268618871659796885667, 3.38334279239515921931826446830, 3.84145774980151186199174462061, 4.61360105668849927513139254339, 5.59922924298147321401904760123, 6.22575304164144736080388268033, 6.88278312250640338906417065967, 8.169229644516153037670053751082, 8.484050776951456418762019193957