L(s) = 1 | + 0.311·2-s + 3-s − 1.90·4-s − 1.52·5-s + 0.311·6-s − 7-s − 1.21·8-s + 9-s − 0.474·10-s − 1.09·11-s − 1.90·12-s − 0.311·14-s − 1.52·15-s + 3.42·16-s + 4.42·17-s + 0.311·18-s + 1.80·19-s + 2.90·20-s − 21-s − 0.341·22-s + 3.80·23-s − 1.21·24-s − 2.67·25-s + 27-s + 1.90·28-s − 0.755·29-s − 0.474·30-s + ⋯ |
L(s) = 1 | + 0.219·2-s + 0.577·3-s − 0.951·4-s − 0.682·5-s + 0.127·6-s − 0.377·7-s − 0.429·8-s + 0.333·9-s − 0.150·10-s − 0.330·11-s − 0.549·12-s − 0.0831·14-s − 0.393·15-s + 0.857·16-s + 1.07·17-s + 0.0733·18-s + 0.414·19-s + 0.649·20-s − 0.218·21-s − 0.0727·22-s + 0.793·23-s − 0.247·24-s − 0.534·25-s + 0.192·27-s + 0.359·28-s − 0.140·29-s − 0.0866·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - 0.311T + 2T^{2} \) |
| 5 | \( 1 + 1.52T + 5T^{2} \) |
| 11 | \( 1 + 1.09T + 11T^{2} \) |
| 17 | \( 1 - 4.42T + 17T^{2} \) |
| 19 | \( 1 - 1.80T + 19T^{2} \) |
| 23 | \( 1 - 3.80T + 23T^{2} \) |
| 29 | \( 1 + 0.755T + 29T^{2} \) |
| 31 | \( 1 - 4.85T + 31T^{2} \) |
| 37 | \( 1 + 5.80T + 37T^{2} \) |
| 41 | \( 1 + 11.3T + 41T^{2} \) |
| 43 | \( 1 - 5.24T + 43T^{2} \) |
| 47 | \( 1 + 2.28T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 0.474T + 59T^{2} \) |
| 61 | \( 1 + 13.0T + 61T^{2} \) |
| 67 | \( 1 + 9.80T + 67T^{2} \) |
| 71 | \( 1 - 13.0T + 71T^{2} \) |
| 73 | \( 1 - 3.47T + 73T^{2} \) |
| 79 | \( 1 + 5.37T + 79T^{2} \) |
| 83 | \( 1 + 13.8T + 83T^{2} \) |
| 89 | \( 1 + 13.1T + 89T^{2} \) |
| 97 | \( 1 + 4.42T + 97T^{2} \) |
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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
−8.191704082940168614541530754681, −7.66969512564356550043956271443, −6.83880109159493490394988219385, −5.76848489580503847188756132254, −5.04330297313452013854579138019, −4.25290270243069877042342930923, −3.41927488060824146090690480056, −2.95977847728260833218594699578, −1.34517993681903708418920664535, 0,
1.34517993681903708418920664535, 2.95977847728260833218594699578, 3.41927488060824146090690480056, 4.25290270243069877042342930923, 5.04330297313452013854579138019, 5.76848489580503847188756132254, 6.83880109159493490394988219385, 7.66969512564356550043956271443, 8.191704082940168614541530754681