| L(s) = 1 | + 9·3-s + 35.3·5-s + 81·9-s + 43.5·11-s − 648.·13-s + 318.·15-s − 598.·17-s + 640.·19-s − 271.·23-s − 1.87e3·25-s + 729·27-s − 5.28e3·29-s + 443.·31-s + 392.·33-s − 8.88e3·37-s − 5.83e3·39-s − 7.48e3·41-s + 3.60e3·43-s + 2.86e3·45-s + 3.01e3·47-s − 5.38e3·51-s − 7.11e3·53-s + 1.54e3·55-s + 5.76e3·57-s − 3.28e4·59-s + 1.91e4·61-s − 2.29e4·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.632·5-s + 0.333·9-s + 0.108·11-s − 1.06·13-s + 0.365·15-s − 0.502·17-s + 0.407·19-s − 0.107·23-s − 0.599·25-s + 0.192·27-s − 1.16·29-s + 0.0828·31-s + 0.0626·33-s − 1.06·37-s − 0.614·39-s − 0.695·41-s + 0.297·43-s + 0.210·45-s + 0.199·47-s − 0.290·51-s − 0.347·53-s + 0.0687·55-s + 0.235·57-s − 1.22·59-s + 0.659·61-s − 0.673·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 9T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 35.3T + 3.12e3T^{2} \) |
| 11 | \( 1 - 43.5T + 1.61e5T^{2} \) |
| 13 | \( 1 + 648.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 598.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 640.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 271.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 5.28e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 443.T + 2.86e7T^{2} \) |
| 37 | \( 1 + 8.88e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 7.48e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 3.60e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 3.01e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 7.11e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.28e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.91e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.49e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.50e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.14e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 5.40e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.21e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 8.90e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.58e4T + 8.58e9T^{2} \) |
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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
−9.537069413418577859384032375537, −8.714999898610115960501227909494, −7.66781451889222459265293861795, −6.90704600287378944120673031670, −5.77709946754771630718600210213, −4.82213163664477157917860142044, −3.65624684845470228314683511353, −2.48925191308599827046257317451, −1.62815905089617252474246930185, 0,
1.62815905089617252474246930185, 2.48925191308599827046257317451, 3.65624684845470228314683511353, 4.82213163664477157917860142044, 5.77709946754771630718600210213, 6.90704600287378944120673031670, 7.66781451889222459265293861795, 8.714999898610115960501227909494, 9.537069413418577859384032375537
