| L(s) = 1 | + 7.24·3-s + 22.1·5-s + 3.68·7-s + 25.5·9-s − 18.5·11-s − 13.7·13-s + 160.·15-s + 46.6·19-s + 26.6·21-s + 178.·23-s + 365.·25-s − 10.8·27-s + 135.·29-s + 115.·31-s − 134.·33-s + 81.4·35-s − 346.·37-s − 99.6·39-s + 255.·41-s − 470.·43-s + 564.·45-s − 233.·47-s − 329.·49-s − 96.0·53-s − 409.·55-s + 338.·57-s + 512.·59-s + ⋯ |
| L(s) = 1 | + 1.39·3-s + 1.97·5-s + 0.198·7-s + 0.944·9-s − 0.507·11-s − 0.293·13-s + 2.76·15-s + 0.563·19-s + 0.277·21-s + 1.61·23-s + 2.92·25-s − 0.0772·27-s + 0.868·29-s + 0.671·31-s − 0.707·33-s + 0.393·35-s − 1.53·37-s − 0.409·39-s + 0.972·41-s − 1.66·43-s + 1.87·45-s − 0.725·47-s − 0.960·49-s − 0.249·53-s − 1.00·55-s + 0.785·57-s + 1.12·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.342437709\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.342437709\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 - 7.24T + 27T^{2} \) |
| 5 | \( 1 - 22.1T + 125T^{2} \) |
| 7 | \( 1 - 3.68T + 343T^{2} \) |
| 11 | \( 1 + 18.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 13.7T + 2.19e3T^{2} \) |
| 19 | \( 1 - 46.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 178.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 135.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 115.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 346.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 255.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 470.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 233.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 96.0T + 1.48e5T^{2} \) |
| 59 | \( 1 - 512.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 642.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 121.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 753.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 155.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 925.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 162.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 265.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 179.T + 9.12e5T^{2} \) |
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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
−8.721691617402958910988172187865, −8.148166505943733226714176618383, −7.05560305544605461329652300811, −6.47101684808963274785273500108, −5.28800977583407701192080262845, −4.93184268699976811957179548171, −3.34938255040175093939448985950, −2.70719742582032913554133005627, −2.02390140587143648046516470438, −1.13058157365654902056238151925,
1.13058157365654902056238151925, 2.02390140587143648046516470438, 2.70719742582032913554133005627, 3.34938255040175093939448985950, 4.93184268699976811957179548171, 5.28800977583407701192080262845, 6.47101684808963274785273500108, 7.05560305544605461329652300811, 8.148166505943733226714176618383, 8.721691617402958910988172187865
