L(s) = 1 | − 3.10·3-s − 5-s + 1.97·7-s + 6.62·9-s + 0.185·11-s + 5.99·13-s + 3.10·15-s − 1.15·17-s + 2.61·19-s − 6.11·21-s − 0.915·23-s + 25-s − 11.2·27-s − 5.92·29-s + 1.52·31-s − 0.576·33-s − 1.97·35-s − 4.46·37-s − 18.6·39-s − 9.05·41-s + 1.79·43-s − 6.62·45-s − 4.47·47-s − 3.10·49-s + 3.59·51-s − 12.0·53-s − 0.185·55-s + ⋯ |
L(s) = 1 | − 1.79·3-s − 0.447·5-s + 0.745·7-s + 2.20·9-s + 0.0560·11-s + 1.66·13-s + 0.800·15-s − 0.280·17-s + 0.599·19-s − 1.33·21-s − 0.190·23-s + 0.200·25-s − 2.16·27-s − 1.10·29-s + 0.274·31-s − 0.100·33-s − 0.333·35-s − 0.733·37-s − 2.97·39-s − 1.41·41-s + 0.273·43-s − 0.987·45-s − 0.653·47-s − 0.444·49-s + 0.503·51-s − 1.65·53-s − 0.0250·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 - T \) |
good | 3 | \( 1 + 3.10T + 3T^{2} \) |
| 7 | \( 1 - 1.97T + 7T^{2} \) |
| 11 | \( 1 - 0.185T + 11T^{2} \) |
| 13 | \( 1 - 5.99T + 13T^{2} \) |
| 17 | \( 1 + 1.15T + 17T^{2} \) |
| 19 | \( 1 - 2.61T + 19T^{2} \) |
| 23 | \( 1 + 0.915T + 23T^{2} \) |
| 29 | \( 1 + 5.92T + 29T^{2} \) |
| 31 | \( 1 - 1.52T + 31T^{2} \) |
| 37 | \( 1 + 4.46T + 37T^{2} \) |
| 41 | \( 1 + 9.05T + 41T^{2} \) |
| 43 | \( 1 - 1.79T + 43T^{2} \) |
| 47 | \( 1 + 4.47T + 47T^{2} \) |
| 53 | \( 1 + 12.0T + 53T^{2} \) |
| 59 | \( 1 + 1.16T + 59T^{2} \) |
| 61 | \( 1 - 9.27T + 61T^{2} \) |
| 67 | \( 1 - 12.9T + 67T^{2} \) |
| 71 | \( 1 + 4.83T + 71T^{2} \) |
| 73 | \( 1 - 3.01T + 73T^{2} \) |
| 79 | \( 1 - 6.21T + 79T^{2} \) |
| 83 | \( 1 + 9.34T + 83T^{2} \) |
| 89 | \( 1 + 17.4T + 89T^{2} \) |
| 97 | \( 1 - 4.82T + 97T^{2} \) |
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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
−7.32282443363823668185372554249, −6.62150477230213218690702723330, −6.11400976082542517224597473710, −5.33589942542712415073373118265, −4.90585819985941117883777172685, −4.04808378741270566556017316862, −3.42710543846794799904925363503, −1.76464726594903658886193192326, −1.13115990512992049218609362628, 0,
1.13115990512992049218609362628, 1.76464726594903658886193192326, 3.42710543846794799904925363503, 4.04808378741270566556017316862, 4.90585819985941117883777172685, 5.33589942542712415073373118265, 6.11400976082542517224597473710, 6.62150477230213218690702723330, 7.32282443363823668185372554249