| L(s) = 1 | − 21.3·2-s + 190.·3-s − 57.2·4-s + 625·5-s − 4.05e3·6-s + 1.21e4·8-s + 1.64e4·9-s − 1.33e4·10-s − 7.98e4·11-s − 1.08e4·12-s − 1.48e5·13-s + 1.18e5·15-s − 2.29e5·16-s − 1.96e5·17-s − 3.50e5·18-s − 7.43e5·19-s − 3.58e4·20-s + 1.70e6·22-s + 1.25e6·23-s + 2.30e6·24-s + 3.90e5·25-s + 3.16e6·26-s − 6.16e5·27-s + 4.19e6·29-s − 2.53e6·30-s − 6.78e6·31-s − 1.32e6·32-s + ⋯ |
| L(s) = 1 | − 0.942·2-s + 1.35·3-s − 0.111·4-s + 0.447·5-s − 1.27·6-s + 1.04·8-s + 0.835·9-s − 0.421·10-s − 1.64·11-s − 0.151·12-s − 1.43·13-s + 0.605·15-s − 0.875·16-s − 0.571·17-s − 0.787·18-s − 1.30·19-s − 0.0500·20-s + 1.55·22-s + 0.937·23-s + 1.41·24-s + 0.200·25-s + 1.35·26-s − 0.223·27-s + 1.10·29-s − 0.570·30-s − 1.32·31-s − 0.222·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.335709453\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.335709453\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 625T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 21.3T + 512T^{2} \) |
| 3 | \( 1 - 190.T + 1.96e4T^{2} \) |
| 11 | \( 1 + 7.98e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.48e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + 1.96e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 7.43e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.25e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 4.19e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 6.78e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 9.11e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.68e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.19e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 5.48e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 3.28e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 7.27e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.26e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 8.96e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.94e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 2.22e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 2.27e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 1.48e7T + 1.86e17T^{2} \) |
| 89 | \( 1 + 4.47e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 6.78e8T + 7.60e17T^{2} \) |
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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
−10.18306327133243174763228969474, −9.327627759326594224169293412709, −8.657062707340141613661500795380, −7.84109814113098251682989450912, −7.11602023002424458840074167344, −5.29388909936135492761886356124, −4.25490167163713234800179354341, −2.56967521334732639926030776139, −2.22049919164464199768366608518, −0.54634292115906621555041427605,
0.54634292115906621555041427605, 2.22049919164464199768366608518, 2.56967521334732639926030776139, 4.25490167163713234800179354341, 5.29388909936135492761886356124, 7.11602023002424458840074167344, 7.84109814113098251682989450912, 8.657062707340141613661500795380, 9.327627759326594224169293412709, 10.18306327133243174763228969474
