L(s) = 1 | + 2-s + 4-s + 5-s − 7-s + 8-s + 10-s − 11-s − 0.813·13-s − 14-s + 16-s − 3.83·17-s + 5.42·19-s + 20-s − 22-s + 5.42·23-s + 25-s − 0.813·26-s − 28-s − 6.61·29-s + 6·31-s + 32-s − 3.83·34-s − 35-s + 8.24·37-s + 5.42·38-s + 40-s + 5.59·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.447·5-s − 0.377·7-s + 0.353·8-s + 0.316·10-s − 0.301·11-s − 0.225·13-s − 0.267·14-s + 0.250·16-s − 0.930·17-s + 1.24·19-s + 0.223·20-s − 0.213·22-s + 1.13·23-s + 0.200·25-s − 0.159·26-s − 0.188·28-s − 1.22·29-s + 1.07·31-s + 0.176·32-s − 0.657·34-s − 0.169·35-s + 1.35·37-s + 0.880·38-s + 0.158·40-s + 0.873·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.403645569\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.403645569\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
good | 13 | \( 1 + 0.813T + 13T^{2} \) |
| 17 | \( 1 + 3.83T + 17T^{2} \) |
| 19 | \( 1 - 5.42T + 19T^{2} \) |
| 23 | \( 1 - 5.42T + 23T^{2} \) |
| 29 | \( 1 + 6.61T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 - 8.24T + 37T^{2} \) |
| 41 | \( 1 - 5.59T + 41T^{2} \) |
| 43 | \( 1 + 3.02T + 43T^{2} \) |
| 47 | \( 1 - 5.02T + 47T^{2} \) |
| 53 | \( 1 + 6.61T + 53T^{2} \) |
| 59 | \( 1 + 10.6T + 59T^{2} \) |
| 61 | \( 1 - 0.978T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 + 14.8T + 71T^{2} \) |
| 73 | \( 1 + 6.20T + 73T^{2} \) |
| 79 | \( 1 - 17.0T + 79T^{2} \) |
| 83 | \( 1 - 8.81T + 83T^{2} \) |
| 89 | \( 1 - 1.18T + 89T^{2} \) |
| 97 | \( 1 + 3.42T + 97T^{2} \) |
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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
−7.60657718774390027113218481676, −7.31417612276527398025533614920, −6.30773720406376424542122735294, −5.94723572993271944544282425128, −4.99152166256193596949802684317, −4.56794340364590913301575095103, −3.47968794812256916267234872759, −2.83715568147452915297999645983, −2.04719756610334211928483101543, −0.846525447533695362917883299068,
0.846525447533695362917883299068, 2.04719756610334211928483101543, 2.83715568147452915297999645983, 3.47968794812256916267234872759, 4.56794340364590913301575095103, 4.99152166256193596949802684317, 5.94723572993271944544282425128, 6.30773720406376424542122735294, 7.31417612276527398025533614920, 7.60657718774390027113218481676