| L(s) = 1 | + 9·3-s − 32.8·5-s + 11.9·7-s + 81·9-s − 21.1·11-s − 812.·13-s − 295.·15-s − 800.·17-s − 1.29e3·19-s + 107.·21-s − 529·23-s − 2.04e3·25-s + 729·27-s − 4.23e3·29-s + 3.12e3·31-s − 190.·33-s − 392.·35-s + 8.80e3·37-s − 7.30e3·39-s + 1.03e4·41-s + 1.04e4·43-s − 2.66e3·45-s + 2.22e4·47-s − 1.66e4·49-s − 7.20e3·51-s − 1.84e4·53-s + 695.·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.588·5-s + 0.0920·7-s + 0.333·9-s − 0.0527·11-s − 1.33·13-s − 0.339·15-s − 0.671·17-s − 0.825·19-s + 0.0531·21-s − 0.208·23-s − 0.654·25-s + 0.192·27-s − 0.935·29-s + 0.583·31-s − 0.0304·33-s − 0.0541·35-s + 1.05·37-s − 0.769·39-s + 0.960·41-s + 0.864·43-s − 0.196·45-s + 1.46·47-s − 0.991·49-s − 0.387·51-s − 0.903·53-s + 0.0310·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.594037087\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.594037087\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 9T \) |
| 23 | \( 1 + 529T \) |
| good | 5 | \( 1 + 32.8T + 3.12e3T^{2} \) |
| 7 | \( 1 - 11.9T + 1.68e4T^{2} \) |
| 11 | \( 1 + 21.1T + 1.61e5T^{2} \) |
| 13 | \( 1 + 812.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 800.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.29e3T + 2.47e6T^{2} \) |
| 29 | \( 1 + 4.23e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.12e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 8.80e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.03e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.04e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.22e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.84e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.63e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.82e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.53e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.94e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.67e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 9.97e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 7.93e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 5.49e3T + 5.58e9T^{2} \) |
| 97 | \( 1 + 6.54e4T + 8.58e9T^{2} \) |
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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
−9.178173193977026860877729583588, −8.156555373879099973786373722184, −7.63034975847197940547694354903, −6.81588857275363351221467766102, −5.73448284475525637051641573478, −4.54086150434933791379065378889, −4.00431097778399449203321088148, −2.73530425115021782691409722715, −2.01274068330647361984134036655, −0.49958944649599673966156473254,
0.49958944649599673966156473254, 2.01274068330647361984134036655, 2.73530425115021782691409722715, 4.00431097778399449203321088148, 4.54086150434933791379065378889, 5.73448284475525637051641573478, 6.81588857275363351221467766102, 7.63034975847197940547694354903, 8.156555373879099973786373722184, 9.178173193977026860877729583588
