| L(s) = 1 | − 1.56e4·5-s − 1.70e5·7-s + 4.56e6·11-s + 7.99e6·13-s − 4.52e7·17-s − 2.50e7·19-s − 1.12e9·23-s + 2.44e8·25-s + 1.59e9·29-s + 3.41e9·31-s + 2.66e9·35-s + 2.50e9·37-s − 6.32e9·41-s + 6.83e10·43-s + 7.28e10·47-s − 6.78e10·49-s + 4.11e10·53-s − 7.12e10·55-s − 3.47e11·59-s + 2.34e11·61-s − 1.24e11·65-s + 3.78e11·67-s − 5.71e11·71-s + 8.27e11·73-s − 7.76e11·77-s + 2.44e12·79-s − 4.67e12·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.547·7-s + 0.776·11-s + 0.459·13-s − 0.454·17-s − 0.122·19-s − 1.58·23-s + 0.199·25-s + 0.496·29-s + 0.691·31-s + 0.244·35-s + 0.160·37-s − 0.208·41-s + 1.64·43-s + 0.985·47-s − 0.700·49-s + 0.255·53-s − 0.347·55-s − 1.07·59-s + 0.583·61-s − 0.205·65-s + 0.510·67-s − 0.529·71-s + 0.639·73-s − 0.424·77-s + 1.13·79-s − 1.56·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 1.56e4T \) |
| good | 7 | \( 1 + 1.70e5T + 9.68e10T^{2} \) |
| 11 | \( 1 - 4.56e6T + 3.45e13T^{2} \) |
| 13 | \( 1 - 7.99e6T + 3.02e14T^{2} \) |
| 17 | \( 1 + 4.52e7T + 9.90e15T^{2} \) |
| 19 | \( 1 + 2.50e7T + 4.20e16T^{2} \) |
| 23 | \( 1 + 1.12e9T + 5.04e17T^{2} \) |
| 29 | \( 1 - 1.59e9T + 1.02e19T^{2} \) |
| 31 | \( 1 - 3.41e9T + 2.44e19T^{2} \) |
| 37 | \( 1 - 2.50e9T + 2.43e20T^{2} \) |
| 41 | \( 1 + 6.32e9T + 9.25e20T^{2} \) |
| 43 | \( 1 - 6.83e10T + 1.71e21T^{2} \) |
| 47 | \( 1 - 7.28e10T + 5.46e21T^{2} \) |
| 53 | \( 1 - 4.11e10T + 2.60e22T^{2} \) |
| 59 | \( 1 + 3.47e11T + 1.04e23T^{2} \) |
| 61 | \( 1 - 2.34e11T + 1.61e23T^{2} \) |
| 67 | \( 1 - 3.78e11T + 5.48e23T^{2} \) |
| 71 | \( 1 + 5.71e11T + 1.16e24T^{2} \) |
| 73 | \( 1 - 8.27e11T + 1.67e24T^{2} \) |
| 79 | \( 1 - 2.44e12T + 4.66e24T^{2} \) |
| 83 | \( 1 + 4.67e12T + 8.87e24T^{2} \) |
| 89 | \( 1 + 9.66e11T + 2.19e25T^{2} \) |
| 97 | \( 1 - 1.18e13T + 6.73e25T^{2} \) |
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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
−9.785979087385831864286589497425, −8.831476825810461247469609652571, −7.86748807660932256416418938193, −6.69240296978012039872515083853, −5.92718434745668335132298875008, −4.41458658578043256522327605725, −3.65450785022310253666699267313, −2.41274856419441983060874383219, −1.10229806877567368693622519018, 0,
1.10229806877567368693622519018, 2.41274856419441983060874383219, 3.65450785022310253666699267313, 4.41458658578043256522327605725, 5.92718434745668335132298875008, 6.69240296978012039872515083853, 7.86748807660932256416418938193, 8.831476825810461247469609652571, 9.785979087385831864286589497425
