| L(s) = 1 | − 7-s − 6·11-s − 6·13-s + 2·17-s − 4·19-s − 2·23-s − 8·29-s + 4·31-s − 6·37-s + 10·41-s − 4·43-s + 4·47-s + 49-s − 4·53-s + 12·59-s + 2·61-s + 12·67-s + 6·71-s + 2·73-s + 6·77-s − 8·79-s − 14·89-s + 6·91-s + 2·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 1.80·11-s − 1.66·13-s + 0.485·17-s − 0.917·19-s − 0.417·23-s − 1.48·29-s + 0.718·31-s − 0.986·37-s + 1.56·41-s − 0.609·43-s + 0.583·47-s + 1/7·49-s − 0.549·53-s + 1.56·59-s + 0.256·61-s + 1.46·67-s + 0.712·71-s + 0.234·73-s + 0.683·77-s − 0.900·79-s − 1.48·89-s + 0.628·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.11311416916240, −13.36969374996456, −12.86858769879293, −12.64209165047376, −12.24722265489013, −11.44712527849511, −11.10596363832511, −10.35089377187772, −10.08402719903012, −9.752982421664239, −9.067153330577485, −8.434146113478725, −7.943218168124145, −7.438442723443876, −7.148179186747639, −6.375096212011331, −5.772026464776095, −5.187605823475153, −4.963509392965850, −4.123959832097786, −3.588489765745931, −2.680584668527482, −2.467049083502390, −1.842344570503515, −0.6003248465398630, 0,
0.6003248465398630, 1.842344570503515, 2.467049083502390, 2.680584668527482, 3.588489765745931, 4.123959832097786, 4.963509392965850, 5.187605823475153, 5.772026464776095, 6.375096212011331, 7.148179186747639, 7.438442723443876, 7.943218168124145, 8.434146113478725, 9.067153330577485, 9.752982421664239, 10.08402719903012, 10.35089377187772, 11.10596363832511, 11.44712527849511, 12.24722265489013, 12.64209165047376, 12.86858769879293, 13.36969374996456, 14.11311416916240
