| L(s) = 1 | − 3-s + 3·9-s + 7·11-s + 3·13-s + 5·17-s − 2·19-s − 2·23-s − 8·27-s − 3·29-s + 16·31-s − 7·33-s + 4·37-s − 3·39-s − 2·41-s + 6·43-s + 3·47-s − 5·51-s − 10·53-s + 2·57-s − 16·59-s − 6·61-s + 8·67-s + 2·69-s + 16·71-s − 12·73-s + 13·79-s + 8·81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 9-s + 2.11·11-s + 0.832·13-s + 1.21·17-s − 0.458·19-s − 0.417·23-s − 1.53·27-s − 0.557·29-s + 2.87·31-s − 1.21·33-s + 0.657·37-s − 0.480·39-s − 0.312·41-s + 0.914·43-s + 0.437·47-s − 0.700·51-s − 1.37·53-s + 0.264·57-s − 2.08·59-s − 0.768·61-s + 0.977·67-s + 0.240·69-s + 1.89·71-s − 1.40·73-s + 1.46·79-s + 8/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 96040000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96040000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.765425130\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.765425130\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 7 T + 26 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 3 T + 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 5 T + 32 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2 T + 14 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 3 T + 52 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 2 T + 50 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 6 T + 62 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 3 T + 22 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 10 T + 98 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + 6 T + 98 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 - 13 T + 126 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 18 T + 226 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 9 T + 8 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.86714723497221161380809548023, −7.60287659806960057404264341069, −6.98489226671069232572602705002, −6.70297286226841982087710512161, −6.49377924569455745301094191249, −6.17963764255314951934948004754, −5.81107162237898039128963092395, −5.74037217576067428776473047705, −4.95906758615692132552918093875, −4.68784507001603419665363480575, −4.39697269276065564031031921679, −3.91028887282140022500451928479, −3.77851466678552415570897398460, −3.42215475923401958058605348395, −2.79882268613991046795181085587, −2.40486744514187607037005108807, −1.62982739048022688618107438862, −1.40034709472953187528692004318, −1.11136159403752732053757160970, −0.50435412087122258433354341545,
0.50435412087122258433354341545, 1.11136159403752732053757160970, 1.40034709472953187528692004318, 1.62982739048022688618107438862, 2.40486744514187607037005108807, 2.79882268613991046795181085587, 3.42215475923401958058605348395, 3.77851466678552415570897398460, 3.91028887282140022500451928479, 4.39697269276065564031031921679, 4.68784507001603419665363480575, 4.95906758615692132552918093875, 5.74037217576067428776473047705, 5.81107162237898039128963092395, 6.17963764255314951934948004754, 6.49377924569455745301094191249, 6.70297286226841982087710512161, 6.98489226671069232572602705002, 7.60287659806960057404264341069, 7.86714723497221161380809548023
