| L(s) = 1 | − 0.614·3-s + 5-s − 2.24·7-s − 2.62·9-s − 5.19·13-s − 0.614·15-s + 3.22·17-s − 7.19·19-s + 1.38·21-s − 5.11·23-s + 25-s + 3.45·27-s − 9.61·29-s − 0.491·31-s − 2.24·35-s + 0.350·37-s + 3.19·39-s + 4.42·41-s − 10.9·43-s − 2.62·45-s + 1.03·47-s − 1.95·49-s − 1.98·51-s + 3.75·53-s + 4.42·57-s − 12.7·59-s + 6.38·61-s + ⋯ |
| L(s) = 1 | − 0.355·3-s + 0.447·5-s − 0.848·7-s − 0.873·9-s − 1.44·13-s − 0.158·15-s + 0.783·17-s − 1.65·19-s + 0.301·21-s − 1.06·23-s + 0.200·25-s + 0.665·27-s − 1.78·29-s − 0.0882·31-s − 0.379·35-s + 0.0576·37-s + 0.511·39-s + 0.690·41-s − 1.67·43-s − 0.390·45-s + 0.150·47-s − 0.279·49-s − 0.278·51-s + 0.515·53-s + 0.586·57-s − 1.65·59-s + 0.818·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4893222609\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4893222609\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 0.614T + 3T^{2} \) |
| 7 | \( 1 + 2.24T + 7T^{2} \) |
| 13 | \( 1 + 5.19T + 13T^{2} \) |
| 17 | \( 1 - 3.22T + 17T^{2} \) |
| 19 | \( 1 + 7.19T + 19T^{2} \) |
| 23 | \( 1 + 5.11T + 23T^{2} \) |
| 29 | \( 1 + 9.61T + 29T^{2} \) |
| 31 | \( 1 + 0.491T + 31T^{2} \) |
| 37 | \( 1 - 0.350T + 37T^{2} \) |
| 41 | \( 1 - 4.42T + 41T^{2} \) |
| 43 | \( 1 + 10.9T + 43T^{2} \) |
| 47 | \( 1 - 1.03T + 47T^{2} \) |
| 53 | \( 1 - 3.75T + 53T^{2} \) |
| 59 | \( 1 + 12.7T + 59T^{2} \) |
| 61 | \( 1 - 6.38T + 61T^{2} \) |
| 67 | \( 1 - 8.44T + 67T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 - 12.0T + 73T^{2} \) |
| 79 | \( 1 - 2.13T + 79T^{2} \) |
| 83 | \( 1 + 10.7T + 83T^{2} \) |
| 89 | \( 1 - 9.90T + 89T^{2} \) |
| 97 | \( 1 + 15.9T + 97T^{2} \) |
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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
−7.75201445733443553530383112322, −6.74425302621874601725908790644, −6.39366095662256935588445144551, −5.57532376217648630660399686495, −5.18463742497707381526634424494, −4.16673942864021978580754812557, −3.39969110657199461396972825497, −2.51676712843654374212084208623, −1.91120387408017757701823902231, −0.31409571975259975733565780637,
0.31409571975259975733565780637, 1.91120387408017757701823902231, 2.51676712843654374212084208623, 3.39969110657199461396972825497, 4.16673942864021978580754812557, 5.18463742497707381526634424494, 5.57532376217648630660399686495, 6.39366095662256935588445144551, 6.74425302621874601725908790644, 7.75201445733443553530383112322
