| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s − 7-s + 8-s + 9-s + 10-s + 11-s + 12-s + 13-s − 14-s + 15-s + 16-s − 17-s + 18-s − 19-s + 20-s − 21-s + 22-s + 23-s + 24-s + 25-s + 26-s + 27-s − 28-s + ⋯ |
| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s − 7-s + 8-s + 9-s + 10-s + 11-s + 12-s + 13-s − 14-s + 15-s + 16-s − 17-s + 18-s − 19-s + 20-s − 21-s + 22-s + 23-s + 24-s + 25-s + 26-s + 27-s − 28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1031 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1031 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(8.632074868\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.632074868\) |
| \(L(1)\) |
\(\approx\) |
\(3.424432297\) |
| \(L(1)\) |
\(\approx\) |
\(3.424432297\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 1031 | \( 1 \) |
| good | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 + T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 + T \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 + T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−21.42543638555137780672971796365, −20.73493804227185617921888243085, −19.947621049242493133390164907167, −19.34899651754004841373857537725, −18.504824767379194072437036309759, −17.272204229819536653149478239499, −16.50804420753397651899959435416, −15.59971809088615957631611434155, −14.98996774495742839982736795264, −14.04075408389827355409628331917, −13.55547200076519402359172975365, −12.95519501725601762435512363994, −12.24938065061276106324961791704, −10.8537807198174235003680097674, −10.28485001826927581931688010107, −9.0666431624907857196483775161, −8.7706391033171722109839017287, −7.13633970817425134081829654227, −6.579648395483394313933279110388, −5.93049129292918572395389638056, −4.61292627816999487508568398384, −3.78059229403184770827031967021, −2.99384159892549382518753739817, −2.11705029197208725078443336089, −1.23567779762651016348635665943,
1.23567779762651016348635665943, 2.11705029197208725078443336089, 2.99384159892549382518753739817, 3.78059229403184770827031967021, 4.61292627816999487508568398384, 5.93049129292918572395389638056, 6.579648395483394313933279110388, 7.13633970817425134081829654227, 8.7706391033171722109839017287, 9.0666431624907857196483775161, 10.28485001826927581931688010107, 10.8537807198174235003680097674, 12.24938065061276106324961791704, 12.95519501725601762435512363994, 13.55547200076519402359172975365, 14.04075408389827355409628331917, 14.98996774495742839982736795264, 15.59971809088615957631611434155, 16.50804420753397651899959435416, 17.272204229819536653149478239499, 18.504824767379194072437036309759, 19.34899651754004841373857537725, 19.947621049242493133390164907167, 20.73493804227185617921888243085, 21.42543638555137780672971796365
