| L(s) = 1 | + 4.47·2-s − 1.90·3-s + 12.0·4-s − 6.52·5-s − 8.52·6-s + 5.22·7-s + 17.8·8-s − 23.3·9-s − 29.1·10-s − 21.1·11-s − 22.8·12-s + 83.4·13-s + 23.3·14-s + 12.4·15-s − 15.9·16-s + 11.3·17-s − 104.·18-s − 7.68·19-s − 78.3·20-s − 9.96·21-s − 94.5·22-s + 153.·23-s − 34.1·24-s − 82.3·25-s + 373.·26-s + 95.9·27-s + 62.7·28-s + ⋯ |
| L(s) = 1 | + 1.58·2-s − 0.366·3-s + 1.50·4-s − 0.583·5-s − 0.579·6-s + 0.282·7-s + 0.791·8-s − 0.865·9-s − 0.923·10-s − 0.579·11-s − 0.550·12-s + 1.78·13-s + 0.446·14-s + 0.214·15-s − 0.249·16-s + 0.161·17-s − 1.36·18-s − 0.0927·19-s − 0.876·20-s − 0.103·21-s − 0.915·22-s + 1.38·23-s − 0.290·24-s − 0.659·25-s + 2.81·26-s + 0.684·27-s + 0.423·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.081377647\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.081377647\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 + 29T \) |
| good | 2 | \( 1 - 4.47T + 8T^{2} \) |
| 3 | \( 1 + 1.90T + 27T^{2} \) |
| 5 | \( 1 + 6.52T + 125T^{2} \) |
| 7 | \( 1 - 5.22T + 343T^{2} \) |
| 11 | \( 1 + 21.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 83.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 11.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 7.68T + 6.85e3T^{2} \) |
| 23 | \( 1 - 153.T + 1.21e4T^{2} \) |
| 31 | \( 1 - 270.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 298.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 184.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 208.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 553.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 321.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 104.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 464.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 745.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 509.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 0.374T + 3.89e5T^{2} \) |
| 79 | \( 1 - 610.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 791.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 342.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 601.T + 9.12e5T^{2} \) |
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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
−16.10697629984900844483167131638, −15.28369458640249990624345109309, −14.03361279256547796029472508924, −13.05408104061301641161936277570, −11.70534672442787407060146472243, −10.99340376908580261400617407287, −8.381493237617127174222673623118, −6.38604761076389158484777629029, −5.07768903673597639646621612413, −3.38297875184629213312704840227,
3.38297875184629213312704840227, 5.07768903673597639646621612413, 6.38604761076389158484777629029, 8.381493237617127174222673623118, 10.99340376908580261400617407287, 11.70534672442787407060146472243, 13.05408104061301641161936277570, 14.03361279256547796029472508924, 15.28369458640249990624345109309, 16.10697629984900844483167131638
