L(s) = 1 | + 4.01·2-s − 8.06·3-s + 8.08·4-s − 14.0·5-s − 32.3·6-s + 0.326·8-s + 38.0·9-s − 56.3·10-s + 53.6·11-s − 65.1·12-s − 64.5·13-s + 113.·15-s − 63.3·16-s + 52.7·17-s + 152.·18-s + 11.4·19-s − 113.·20-s + 215.·22-s − 98.9·23-s − 2.63·24-s + 72.5·25-s − 258.·26-s − 89.1·27-s + 42.5·29-s + 454.·30-s + 67.2·31-s − 256.·32-s + ⋯ |
L(s) = 1 | + 1.41·2-s − 1.55·3-s + 1.01·4-s − 1.25·5-s − 2.20·6-s + 0.0144·8-s + 1.40·9-s − 1.78·10-s + 1.47·11-s − 1.56·12-s − 1.37·13-s + 1.95·15-s − 0.989·16-s + 0.753·17-s + 1.99·18-s + 0.137·19-s − 1.26·20-s + 2.08·22-s − 0.896·23-s − 0.0223·24-s + 0.580·25-s − 1.95·26-s − 0.635·27-s + 0.272·29-s + 2.76·30-s + 0.389·31-s − 1.41·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
good | 2 | \( 1 - 4.01T + 8T^{2} \) |
| 3 | \( 1 + 8.06T + 27T^{2} \) |
| 5 | \( 1 + 14.0T + 125T^{2} \) |
| 11 | \( 1 - 53.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 64.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 52.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 11.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 98.9T + 1.21e4T^{2} \) |
| 29 | \( 1 - 42.5T + 2.43e4T^{2} \) |
| 31 | \( 1 - 67.2T + 2.97e4T^{2} \) |
| 37 | \( 1 - 428.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 45.7T + 6.89e4T^{2} \) |
| 43 | \( 1 - 196.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 568.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 151.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 171.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 5.02T + 2.26e5T^{2} \) |
| 67 | \( 1 + 196.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 79.6T + 3.57e5T^{2} \) |
| 73 | \( 1 + 719.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 241.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 421.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 628.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.68e3T + 9.12e5T^{2} \) |
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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
−7.78586593157593944402058633648, −7.16673769384919020078092162102, −6.35296587144060600547123433878, −5.80712999657975721876613504039, −4.94324118926201770209397371290, −4.26244011005207298551946282666, −3.86942142653809842715780167831, −2.63667008544041433101574616468, −0.995854883304714586498732627777, 0,
0.995854883304714586498732627777, 2.63667008544041433101574616468, 3.86942142653809842715780167831, 4.26244011005207298551946282666, 4.94324118926201770209397371290, 5.80712999657975721876613504039, 6.35296587144060600547123433878, 7.16673769384919020078092162102, 7.78586593157593944402058633648