| L(s) = 1 | − 1.44·3-s − 9.31·5-s + 22.5·7-s − 24.9·9-s + 9.73·13-s + 13.4·15-s + 132.·17-s − 29.8·19-s − 32.6·21-s − 175.·23-s − 38.2·25-s + 75.0·27-s − 74.0·29-s − 29.2·31-s − 209.·35-s − 85.4·37-s − 14.0·39-s + 282.·41-s + 212.·43-s + 231.·45-s − 377.·47-s + 165.·49-s − 191.·51-s + 717.·53-s + 43.0·57-s + 57.5·59-s + 146.·61-s + ⋯ |
| L(s) = 1 | − 0.278·3-s − 0.832·5-s + 1.21·7-s − 0.922·9-s + 0.207·13-s + 0.231·15-s + 1.89·17-s − 0.359·19-s − 0.338·21-s − 1.58·23-s − 0.306·25-s + 0.535·27-s − 0.473·29-s − 0.169·31-s − 1.01·35-s − 0.379·37-s − 0.0578·39-s + 1.07·41-s + 0.754·43-s + 0.768·45-s − 1.17·47-s + 0.482·49-s − 0.526·51-s + 1.85·53-s + 0.100·57-s + 0.126·59-s + 0.307·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.511811039\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.511811039\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 1.44T + 27T^{2} \) |
| 5 | \( 1 + 9.31T + 125T^{2} \) |
| 7 | \( 1 - 22.5T + 343T^{2} \) |
| 13 | \( 1 - 9.73T + 2.19e3T^{2} \) |
| 17 | \( 1 - 132.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 29.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 175.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 74.0T + 2.43e4T^{2} \) |
| 31 | \( 1 + 29.2T + 2.97e4T^{2} \) |
| 37 | \( 1 + 85.4T + 5.06e4T^{2} \) |
| 41 | \( 1 - 282.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 212.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 377.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 717.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 57.5T + 2.05e5T^{2} \) |
| 61 | \( 1 - 146.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 792.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 8.64T + 3.57e5T^{2} \) |
| 73 | \( 1 + 973.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 732.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 587.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 826.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 301.T + 9.12e5T^{2} \) |
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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
−9.697630596016589165858726112666, −8.463398639742628225002946118102, −8.041592375905173104054132768319, −7.37228910999860634584296258439, −5.95348767823940795952284672269, −5.39497634160372361133843129677, −4.27463799161628947076197862150, −3.41988393666474196988483963420, −2.01520003214391992933168864841, −0.67021737375667726435793070160,
0.67021737375667726435793070160, 2.01520003214391992933168864841, 3.41988393666474196988483963420, 4.27463799161628947076197862150, 5.39497634160372361133843129677, 5.95348767823940795952284672269, 7.37228910999860634584296258439, 8.041592375905173104054132768319, 8.463398639742628225002946118102, 9.697630596016589165858726112666
