| L(s) = 1 | + 3-s − 5-s + 2.60·7-s + 9-s + 4.60·11-s + 4.60·13-s − 15-s + 2·17-s + 19-s + 2.60·21-s + 2·23-s + 25-s + 27-s + 2.60·29-s − 4·31-s + 4.60·33-s − 2.60·35-s + 3.39·37-s + 4.60·39-s + 6.60·41-s − 10.6·43-s − 45-s − 6·47-s − 0.211·49-s + 2·51-s − 4.60·55-s + 57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.984·7-s + 0.333·9-s + 1.38·11-s + 1.27·13-s − 0.258·15-s + 0.485·17-s + 0.229·19-s + 0.568·21-s + 0.417·23-s + 0.200·25-s + 0.192·27-s + 0.483·29-s − 0.718·31-s + 0.801·33-s − 0.440·35-s + 0.558·37-s + 0.737·39-s + 1.03·41-s − 1.61·43-s − 0.149·45-s − 0.875·47-s − 0.0301·49-s + 0.280·51-s − 0.621·55-s + 0.132·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.184803204\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.184803204\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 - 2.60T + 7T^{2} \) |
| 11 | \( 1 - 4.60T + 11T^{2} \) |
| 13 | \( 1 - 4.60T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 23 | \( 1 - 2T + 23T^{2} \) |
| 29 | \( 1 - 2.60T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 3.39T + 37T^{2} \) |
| 41 | \( 1 - 6.60T + 41T^{2} \) |
| 43 | \( 1 + 10.6T + 43T^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 5.21T + 59T^{2} \) |
| 61 | \( 1 + 7.21T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 - 9.21T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 11.2T + 83T^{2} \) |
| 89 | \( 1 - 6.60T + 89T^{2} \) |
| 97 | \( 1 - 16.6T + 97T^{2} \) |
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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
−8.129990907965740530384268872385, −7.939042342843346260053904486206, −6.87653934196805019528345719844, −6.30714011047980416701707001899, −5.28685499755853746652370235034, −4.43075586664445996428343849319, −3.76659040958673011292069100696, −3.08656680534505380166577694491, −1.71791492910901594510364452790, −1.10860880089444067618322377423,
1.10860880089444067618322377423, 1.71791492910901594510364452790, 3.08656680534505380166577694491, 3.76659040958673011292069100696, 4.43075586664445996428343849319, 5.28685499755853746652370235034, 6.30714011047980416701707001899, 6.87653934196805019528345719844, 7.939042342843346260053904486206, 8.129990907965740530384268872385
