| L(s) = 1 | + 9·3-s − 37.5·5-s + 76.4·7-s + 81·9-s + 121·11-s + 169.·13-s − 337.·15-s − 0.875·17-s + 817.·19-s + 688.·21-s − 749.·23-s − 1.71e3·25-s + 729·27-s + 6.04e3·29-s + 1.47e3·31-s + 1.08e3·33-s − 2.86e3·35-s − 1.58e4·37-s + 1.52e3·39-s − 7.62e3·41-s + 1.82e4·43-s − 3.03e3·45-s + 1.28e4·47-s − 1.09e4·49-s − 7.88·51-s + 2.17e4·53-s − 4.54e3·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.671·5-s + 0.589·7-s + 0.333·9-s + 0.301·11-s + 0.278·13-s − 0.387·15-s − 0.000735·17-s + 0.519·19-s + 0.340·21-s − 0.295·23-s − 0.549·25-s + 0.192·27-s + 1.33·29-s + 0.275·31-s + 0.174·33-s − 0.395·35-s − 1.90·37-s + 0.160·39-s − 0.708·41-s + 1.50·43-s − 0.223·45-s + 0.850·47-s − 0.651·49-s − 0.000424·51-s + 1.06·53-s − 0.202·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.669250645\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.669250645\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 9T \) |
| 11 | \( 1 - 121T \) |
| good | 5 | \( 1 + 37.5T + 3.12e3T^{2} \) |
| 7 | \( 1 - 76.4T + 1.68e4T^{2} \) |
| 13 | \( 1 - 169.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 0.875T + 1.41e6T^{2} \) |
| 19 | \( 1 - 817.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 749.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 6.04e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.47e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.58e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 7.62e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.82e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.28e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.17e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.16e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.40e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.69e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.75e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 8.04e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.21e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.19e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.46e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.38e5T + 8.58e9T^{2} \) |
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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
−10.06051260427517181603584926515, −8.997701593641764150467422808976, −8.264256114169355541554967252134, −7.54832101013523363300282006312, −6.57506374178032371003289668753, −5.26598464874052085670741899365, −4.21077497018825057696912854541, −3.35693388146235237530308778915, −2.05443952216761415302541315625, −0.810835862087451198760653308438,
0.810835862087451198760653308438, 2.05443952216761415302541315625, 3.35693388146235237530308778915, 4.21077497018825057696912854541, 5.26598464874052085670741899365, 6.57506374178032371003289668753, 7.54832101013523363300282006312, 8.264256114169355541554967252134, 8.997701593641764150467422808976, 10.06051260427517181603584926515
