| L(s) = 1 | − 5-s + 7-s − 5·11-s + 17-s − 6·19-s − 6·23-s + 25-s + 9·29-s + 4·31-s − 35-s − 2·37-s − 4·41-s + 10·43-s − 47-s + 49-s − 4·53-s + 5·55-s − 8·59-s − 8·61-s − 12·67-s + 8·71-s − 2·73-s − 5·77-s + 13·79-s − 4·83-s − 85-s + 4·89-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.377·7-s − 1.50·11-s + 0.242·17-s − 1.37·19-s − 1.25·23-s + 1/5·25-s + 1.67·29-s + 0.718·31-s − 0.169·35-s − 0.328·37-s − 0.624·41-s + 1.52·43-s − 0.145·47-s + 1/7·49-s − 0.549·53-s + 0.674·55-s − 1.04·59-s − 1.02·61-s − 1.46·67-s + 0.949·71-s − 0.234·73-s − 0.569·77-s + 1.46·79-s − 0.439·83-s − 0.108·85-s + 0.423·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 212940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 212940 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8616839449\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8616839449\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 11 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−12.88513997140703, −12.50186207421059, −12.12879338642448, −11.69811928306624, −11.02295317677134, −10.58861882708040, −10.32789375126937, −9.937771182919254, −9.082555183062187, −8.710477364410921, −8.128877822166809, −7.819936880813398, −7.549149719314070, −6.688859030049348, −6.251572700765978, −5.842579493258570, −5.086253966752510, −4.648736869288060, −4.342318377198110, −3.585091312486331, −2.970795474261726, −2.435522852831319, −1.952039482322335, −1.096133519358092, −0.2746415559780871,
0.2746415559780871, 1.096133519358092, 1.952039482322335, 2.435522852831319, 2.970795474261726, 3.585091312486331, 4.342318377198110, 4.648736869288060, 5.086253966752510, 5.842579493258570, 6.251572700765978, 6.688859030049348, 7.549149719314070, 7.819936880813398, 8.128877822166809, 8.710477364410921, 9.082555183062187, 9.937771182919254, 10.32789375126937, 10.58861882708040, 11.02295317677134, 11.69811928306624, 12.12879338642448, 12.50186207421059, 12.88513997140703
