L(s) = 1 | − 0.189·3-s + 1.65·7-s − 2.96·9-s − 0.0759·11-s − 1.24·13-s + 5.17·17-s − 0.792·19-s − 0.313·21-s + 23-s + 1.12·27-s − 2.85·29-s − 8.90·31-s + 0.0143·33-s + 6.92·37-s + 0.235·39-s + 4.64·41-s + 1.75·43-s − 10.0·47-s − 4.25·49-s − 0.980·51-s − 11.1·53-s + 0.150·57-s + 12.4·59-s − 12.0·61-s − 4.91·63-s + 6.96·67-s − 0.189·69-s + ⋯ |
L(s) = 1 | − 0.109·3-s + 0.626·7-s − 0.988·9-s − 0.0228·11-s − 0.344·13-s + 1.25·17-s − 0.181·19-s − 0.0684·21-s + 0.208·23-s + 0.217·27-s − 0.529·29-s − 1.59·31-s + 0.00250·33-s + 1.13·37-s + 0.0376·39-s + 0.725·41-s + 0.267·43-s − 1.45·47-s − 0.607·49-s − 0.137·51-s − 1.52·53-s + 0.0198·57-s + 1.62·59-s − 1.53·61-s − 0.618·63-s + 0.850·67-s − 0.0227·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 3 | \( 1 + 0.189T + 3T^{2} \) |
| 7 | \( 1 - 1.65T + 7T^{2} \) |
| 11 | \( 1 + 0.0759T + 11T^{2} \) |
| 13 | \( 1 + 1.24T + 13T^{2} \) |
| 17 | \( 1 - 5.17T + 17T^{2} \) |
| 19 | \( 1 + 0.792T + 19T^{2} \) |
| 29 | \( 1 + 2.85T + 29T^{2} \) |
| 31 | \( 1 + 8.90T + 31T^{2} \) |
| 37 | \( 1 - 6.92T + 37T^{2} \) |
| 41 | \( 1 - 4.64T + 41T^{2} \) |
| 43 | \( 1 - 1.75T + 43T^{2} \) |
| 47 | \( 1 + 10.0T + 47T^{2} \) |
| 53 | \( 1 + 11.1T + 53T^{2} \) |
| 59 | \( 1 - 12.4T + 59T^{2} \) |
| 61 | \( 1 + 12.0T + 61T^{2} \) |
| 67 | \( 1 - 6.96T + 67T^{2} \) |
| 71 | \( 1 - 7.71T + 71T^{2} \) |
| 73 | \( 1 + 7.49T + 73T^{2} \) |
| 79 | \( 1 - 15.6T + 79T^{2} \) |
| 83 | \( 1 + 6.68T + 83T^{2} \) |
| 89 | \( 1 + 3.03T + 89T^{2} \) |
| 97 | \( 1 + 3.21T + 97T^{2} \) |
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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
−7.60917738978960033849192380524, −6.65524285074550254937581965511, −5.88865745240992511900924655833, −5.34429960029813289629293074317, −4.75187593124288271146383952951, −3.75759168588257445022578100305, −3.06842808519690024442715866295, −2.19119933756943432345763921704, −1.23955647603215255479049019914, 0,
1.23955647603215255479049019914, 2.19119933756943432345763921704, 3.06842808519690024442715866295, 3.75759168588257445022578100305, 4.75187593124288271146383952951, 5.34429960029813289629293074317, 5.88865745240992511900924655833, 6.65524285074550254937581965511, 7.60917738978960033849192380524