L(s) = 1 | − 2.57·2-s − 2.27·3-s + 4.61·4-s + 5.84·6-s + 2.96·7-s − 6.71·8-s + 2.16·9-s + 1.65·11-s − 10.4·12-s − 4.85·13-s − 7.62·14-s + 8.05·16-s + 4.10·17-s − 5.57·18-s + 7.52·19-s − 6.74·21-s − 4.25·22-s + 3.98·23-s + 15.2·24-s + 12.4·26-s + 1.88·27-s + 13.6·28-s − 7.13·29-s − 9.98·31-s − 7.27·32-s − 3.76·33-s − 10.5·34-s + ⋯ |
L(s) = 1 | − 1.81·2-s − 1.31·3-s + 2.30·4-s + 2.38·6-s + 1.12·7-s − 2.37·8-s + 0.723·9-s + 0.499·11-s − 3.02·12-s − 1.34·13-s − 2.03·14-s + 2.01·16-s + 0.994·17-s − 1.31·18-s + 1.72·19-s − 1.47·21-s − 0.907·22-s + 0.831·23-s + 3.11·24-s + 2.44·26-s + 0.363·27-s + 2.58·28-s − 1.32·29-s − 1.79·31-s − 1.28·32-s − 0.655·33-s − 1.80·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5678841036\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5678841036\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 241 | \( 1 - T \) |
good | 2 | \( 1 + 2.57T + 2T^{2} \) |
| 3 | \( 1 + 2.27T + 3T^{2} \) |
| 7 | \( 1 - 2.96T + 7T^{2} \) |
| 11 | \( 1 - 1.65T + 11T^{2} \) |
| 13 | \( 1 + 4.85T + 13T^{2} \) |
| 17 | \( 1 - 4.10T + 17T^{2} \) |
| 19 | \( 1 - 7.52T + 19T^{2} \) |
| 23 | \( 1 - 3.98T + 23T^{2} \) |
| 29 | \( 1 + 7.13T + 29T^{2} \) |
| 31 | \( 1 + 9.98T + 31T^{2} \) |
| 37 | \( 1 - 5.09T + 37T^{2} \) |
| 41 | \( 1 - 10.2T + 41T^{2} \) |
| 43 | \( 1 + 11.5T + 43T^{2} \) |
| 47 | \( 1 - 9.37T + 47T^{2} \) |
| 53 | \( 1 - 4.15T + 53T^{2} \) |
| 59 | \( 1 + 8.26T + 59T^{2} \) |
| 61 | \( 1 - 8.26T + 61T^{2} \) |
| 67 | \( 1 + 10.4T + 67T^{2} \) |
| 71 | \( 1 + 0.747T + 71T^{2} \) |
| 73 | \( 1 - 6.80T + 73T^{2} \) |
| 79 | \( 1 - 13.0T + 79T^{2} \) |
| 83 | \( 1 - 7.04T + 83T^{2} \) |
| 89 | \( 1 - 12.6T + 89T^{2} \) |
| 97 | \( 1 - 10.2T + 97T^{2} \) |
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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
−7.80636817179136848040224264932, −7.52478129702692134437525176593, −7.06582357273239660139614701164, −6.02866632575309086866901097979, −5.37983384802691761826966157902, −4.85035253099442057359613809701, −3.42707687427275520159550477104, −2.24555871831578368768071400324, −1.33251783370258114598352433678, −0.62973196436377668508068197197,
0.62973196436377668508068197197, 1.33251783370258114598352433678, 2.24555871831578368768071400324, 3.42707687427275520159550477104, 4.85035253099442057359613809701, 5.37983384802691761826966157902, 6.02866632575309086866901097979, 7.06582357273239660139614701164, 7.52478129702692134437525176593, 7.80636817179136848040224264932