| L(s) = 1 | − 2·3-s − 3·5-s − 7-s + 9-s + 6·13-s + 6·15-s − 6·19-s + 2·21-s + 23-s + 4·25-s + 4·27-s + 2·31-s + 3·35-s − 10·37-s − 12·39-s + 5·41-s + 11·43-s − 3·45-s − 2·47-s + 49-s − 8·53-s + 12·57-s + 6·59-s − 63-s − 18·65-s + 12·67-s − 2·69-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1.34·5-s − 0.377·7-s + 1/3·9-s + 1.66·13-s + 1.54·15-s − 1.37·19-s + 0.436·21-s + 0.208·23-s + 4/5·25-s + 0.769·27-s + 0.359·31-s + 0.507·35-s − 1.64·37-s − 1.92·39-s + 0.780·41-s + 1.67·43-s − 0.447·45-s − 0.291·47-s + 1/7·49-s − 1.09·53-s + 1.58·57-s + 0.781·59-s − 0.125·63-s − 2.23·65-s + 1.46·67-s − 0.240·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 155848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155848 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8608459288\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8608459288\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−12.96436805050439, −12.77671562534797, −12.33480279829810, −11.77252971189968, −11.35679691886474, −11.01353552745182, −10.67005474358926, −10.24708788957892, −9.442098194490710, −8.768646884421643, −8.511103755734383, −8.034007015835747, −7.390437160843753, −6.773412247720106, −6.474183273912254, −5.900225948668055, −5.526372862091061, −4.716225906127340, −4.316621244719817, −3.749980304424939, −3.373458770275226, −2.586960373257782, −1.705974075595864, −0.8487765423445628, −0.4015027000737255,
0.4015027000737255, 0.8487765423445628, 1.705974075595864, 2.586960373257782, 3.373458770275226, 3.749980304424939, 4.316621244719817, 4.716225906127340, 5.526372862091061, 5.900225948668055, 6.474183273912254, 6.773412247720106, 7.390437160843753, 8.034007015835747, 8.511103755734383, 8.768646884421643, 9.442098194490710, 10.24708788957892, 10.67005474358926, 11.01353552745182, 11.35679691886474, 11.77252971189968, 12.33480279829810, 12.77671562534797, 12.96436805050439
