| L(s) = 1 | − 1.35·3-s + 5-s + 3.00·7-s − 1.17·9-s + 5.69·13-s − 1.35·15-s + 4.70·17-s + 3.69·19-s − 4.06·21-s + 6.83·23-s + 25-s + 5.64·27-s + 10.6·29-s + 3.50·31-s + 3.00·35-s + 5.79·37-s − 7.69·39-s + 8.39·41-s − 2.80·43-s − 1.17·45-s − 5.14·47-s + 2.04·49-s − 6.35·51-s − 12.2·53-s − 4.99·57-s − 3.30·59-s + 10.7·61-s + ⋯ |
| L(s) = 1 | − 0.779·3-s + 0.447·5-s + 1.13·7-s − 0.391·9-s + 1.58·13-s − 0.348·15-s + 1.14·17-s + 0.848·19-s − 0.886·21-s + 1.42·23-s + 0.200·25-s + 1.08·27-s + 1.98·29-s + 0.629·31-s + 0.508·35-s + 0.952·37-s − 1.23·39-s + 1.31·41-s − 0.428·43-s − 0.175·45-s − 0.750·47-s + 0.291·49-s − 0.889·51-s − 1.67·53-s − 0.661·57-s − 0.430·59-s + 1.38·61-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
| \(L(1)\) |
\(\approx\) |
\(2.709342080\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.709342080\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.35T + 3T^{2} \) |
| 7 | \( 1 - 3.00T + 7T^{2} \) |
| 13 | \( 1 - 5.69T + 13T^{2} \) |
| 17 | \( 1 - 4.70T + 17T^{2} \) |
| 19 | \( 1 - 3.69T + 19T^{2} \) |
| 23 | \( 1 - 6.83T + 23T^{2} \) |
| 29 | \( 1 - 10.6T + 29T^{2} \) |
| 31 | \( 1 - 3.50T + 31T^{2} \) |
| 37 | \( 1 - 5.79T + 37T^{2} \) |
| 41 | \( 1 - 8.39T + 41T^{2} \) |
| 43 | \( 1 + 2.80T + 43T^{2} \) |
| 47 | \( 1 + 5.14T + 47T^{2} \) |
| 53 | \( 1 + 12.2T + 53T^{2} \) |
| 59 | \( 1 + 3.30T + 59T^{2} \) |
| 61 | \( 1 - 10.7T + 61T^{2} \) |
| 67 | \( 1 + 11.5T + 67T^{2} \) |
| 71 | \( 1 + 2.05T + 71T^{2} \) |
| 73 | \( 1 - 3.29T + 73T^{2} \) |
| 79 | \( 1 - 4.67T + 79T^{2} \) |
| 83 | \( 1 - 12.5T + 83T^{2} \) |
| 89 | \( 1 + 4.32T + 89T^{2} \) |
| 97 | \( 1 - 2.94T + 97T^{2} \) |
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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
−7.967673559569418405984452472621, −6.75047496678155347327184458000, −6.27127487597631185003240413011, −5.58724116580104958295893792909, −5.04722076105367088556703387346, −4.47106539692592891815126923256, −3.30895275578112477486184098865, −2.70614501804200303072016384044, −1.19496943320718767919542530223, −1.09071032803553296888731433467,
1.09071032803553296888731433467, 1.19496943320718767919542530223, 2.70614501804200303072016384044, 3.30895275578112477486184098865, 4.47106539692592891815126923256, 5.04722076105367088556703387346, 5.58724116580104958295893792909, 6.27127487597631185003240413011, 6.75047496678155347327184458000, 7.967673559569418405984452472621
