| L(s) = 1 | − 4·11-s − 4·13-s − 2·17-s + 4·19-s + 8·23-s − 5·25-s + 8·29-s + 4·31-s − 4·37-s + 6·41-s + 4·43-s − 8·47-s + 8·53-s − 12·59-s − 12·61-s + 12·67-s − 8·71-s + 6·73-s − 4·79-s − 4·83-s − 6·89-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 1.20·11-s − 1.10·13-s − 0.485·17-s + 0.917·19-s + 1.66·23-s − 25-s + 1.48·29-s + 0.718·31-s − 0.657·37-s + 0.937·41-s + 0.609·43-s − 1.16·47-s + 1.09·53-s − 1.56·59-s − 1.53·61-s + 1.46·67-s − 0.949·71-s + 0.702·73-s − 0.450·79-s − 0.439·83-s − 0.635·89-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−13.98984387977040, −13.35937053441517, −12.93510773636571, −12.35635150762648, −12.06760226653256, −11.40186658982277, −10.94679654722648, −10.45222491302302, −9.947215885339840, −9.538243644111688, −9.005403443078937, −8.360657625967805, −7.921939187664656, −7.327618695379082, −7.073803816484947, −6.322842489844212, −5.763357289054612, −5.060574161377061, −4.856053352506519, −4.270004459181468, −3.354725071035031, −2.750918675689893, −2.541380109422199, −1.589044813326043, −0.7933906495621620, 0,
0.7933906495621620, 1.589044813326043, 2.541380109422199, 2.750918675689893, 3.354725071035031, 4.270004459181468, 4.856053352506519, 5.060574161377061, 5.763357289054612, 6.322842489844212, 7.073803816484947, 7.327618695379082, 7.921939187664656, 8.360657625967805, 9.005403443078937, 9.538243644111688, 9.947215885339840, 10.45222491302302, 10.94679654722648, 11.40186658982277, 12.06760226653256, 12.35635150762648, 12.93510773636571, 13.35937053441517, 13.98984387977040
