L(s) = 1 | − 1.98·2-s − 3-s + 1.94·4-s − 5-s + 1.98·6-s + 4.82·7-s + 0.114·8-s + 9-s + 1.98·10-s − 3.26·11-s − 1.94·12-s + 4.27·13-s − 9.58·14-s + 15-s − 4.11·16-s + 0.810·17-s − 1.98·18-s + 1.34·19-s − 1.94·20-s − 4.82·21-s + 6.47·22-s + 7.26·23-s − 0.114·24-s + 25-s − 8.49·26-s − 27-s + 9.38·28-s + ⋯ |
L(s) = 1 | − 1.40·2-s − 0.577·3-s + 0.971·4-s − 0.447·5-s + 0.810·6-s + 1.82·7-s + 0.0403·8-s + 0.333·9-s + 0.627·10-s − 0.983·11-s − 0.560·12-s + 1.18·13-s − 2.56·14-s + 0.258·15-s − 1.02·16-s + 0.196·17-s − 0.468·18-s + 0.307·19-s − 0.434·20-s − 1.05·21-s + 1.38·22-s + 1.51·23-s − 0.0232·24-s + 0.200·25-s − 1.66·26-s − 0.192·27-s + 1.77·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 + T \) |
good | 2 | \( 1 + 1.98T + 2T^{2} \) |
| 7 | \( 1 - 4.82T + 7T^{2} \) |
| 11 | \( 1 + 3.26T + 11T^{2} \) |
| 13 | \( 1 - 4.27T + 13T^{2} \) |
| 17 | \( 1 - 0.810T + 17T^{2} \) |
| 19 | \( 1 - 1.34T + 19T^{2} \) |
| 23 | \( 1 - 7.26T + 23T^{2} \) |
| 29 | \( 1 - 3.20T + 29T^{2} \) |
| 31 | \( 1 + 7.76T + 31T^{2} \) |
| 37 | \( 1 + 3.62T + 37T^{2} \) |
| 41 | \( 1 + 11.7T + 41T^{2} \) |
| 43 | \( 1 + 3.17T + 43T^{2} \) |
| 47 | \( 1 + 12.4T + 47T^{2} \) |
| 53 | \( 1 + 7.31T + 53T^{2} \) |
| 59 | \( 1 - 0.540T + 59T^{2} \) |
| 61 | \( 1 - 5.50T + 61T^{2} \) |
| 67 | \( 1 - 6.91T + 67T^{2} \) |
| 71 | \( 1 + 7.94T + 71T^{2} \) |
| 73 | \( 1 + 9.45T + 73T^{2} \) |
| 79 | \( 1 + 6.39T + 79T^{2} \) |
| 83 | \( 1 + 0.821T + 83T^{2} \) |
| 89 | \( 1 + 18.5T + 89T^{2} \) |
| 97 | \( 1 - 0.781T + 97T^{2} \) |
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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
−7.988171213956162472583139695509, −7.23412134520196589011603964129, −6.69358267739344423511605858893, −5.32290115492874836656836134193, −5.10335332105533825182932292629, −4.19260734716437584290494269167, −3.03804334413381803591290867875, −1.66100020022205023925008444808, −1.28300765255200130040836697150, 0,
1.28300765255200130040836697150, 1.66100020022205023925008444808, 3.03804334413381803591290867875, 4.19260734716437584290494269167, 5.10335332105533825182932292629, 5.32290115492874836656836134193, 6.69358267739344423511605858893, 7.23412134520196589011603964129, 7.988171213956162472583139695509