| L(s) = 1 | + 3-s + 5-s + 1.56·7-s + 9-s + 1.56·11-s + 2·13-s + 15-s + 5.12·17-s − 4.68·19-s + 1.56·21-s − 5.56·23-s + 25-s + 27-s − 1.12·29-s − 31-s + 1.56·33-s + 1.56·35-s + 5.12·37-s + 2·39-s − 1.12·41-s + 7.80·43-s + 45-s + 3.12·47-s − 4.56·49-s + 5.12·51-s + 11.5·53-s + 1.56·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.590·7-s + 0.333·9-s + 0.470·11-s + 0.554·13-s + 0.258·15-s + 1.24·17-s − 1.07·19-s + 0.340·21-s − 1.15·23-s + 0.200·25-s + 0.192·27-s − 0.208·29-s − 0.179·31-s + 0.271·33-s + 0.263·35-s + 0.842·37-s + 0.320·39-s − 0.175·41-s + 1.19·43-s + 0.149·45-s + 0.455·47-s − 0.651·49-s + 0.717·51-s + 1.58·53-s + 0.210·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.419638905\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.419638905\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| good | 7 | \( 1 - 1.56T + 7T^{2} \) |
| 11 | \( 1 - 1.56T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 5.12T + 17T^{2} \) |
| 19 | \( 1 + 4.68T + 19T^{2} \) |
| 23 | \( 1 + 5.56T + 23T^{2} \) |
| 29 | \( 1 + 1.12T + 29T^{2} \) |
| 37 | \( 1 - 5.12T + 37T^{2} \) |
| 41 | \( 1 + 1.12T + 41T^{2} \) |
| 43 | \( 1 - 7.80T + 43T^{2} \) |
| 47 | \( 1 - 3.12T + 47T^{2} \) |
| 53 | \( 1 - 11.5T + 53T^{2} \) |
| 59 | \( 1 - 4.87T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 + 9.36T + 67T^{2} \) |
| 71 | \( 1 + 4.68T + 71T^{2} \) |
| 73 | \( 1 - 9.80T + 73T^{2} \) |
| 79 | \( 1 - 16.6T + 79T^{2} \) |
| 83 | \( 1 - 2.24T + 83T^{2} \) |
| 89 | \( 1 + 1.31T + 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−7.953679186914090726885351048719, −7.36637808857125504513710442176, −6.40942840968480926456271871619, −5.88835378692802499152407847734, −5.09841851498645150460449934458, −4.12186358114590342152312771630, −3.68984532520236152109015142385, −2.56178389189696991677469835605, −1.86377409912771564889771108507, −0.953577815886392368258272183956,
0.953577815886392368258272183956, 1.86377409912771564889771108507, 2.56178389189696991677469835605, 3.68984532520236152109015142385, 4.12186358114590342152312771630, 5.09841851498645150460449934458, 5.88835378692802499152407847734, 6.40942840968480926456271871619, 7.36637808857125504513710442176, 7.953679186914090726885351048719
