| L(s) = 1 | + 3-s − 5-s + 7-s + 9-s + 5·11-s + 5·13-s − 15-s + 17-s + 5·19-s + 21-s − 23-s − 4·25-s + 27-s + 6·29-s − 6·31-s + 5·33-s − 35-s − 4·37-s + 5·39-s + 7·41-s + 7·43-s − 45-s + 6·47-s + 49-s + 51-s − 6·53-s − 5·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 1.50·11-s + 1.38·13-s − 0.258·15-s + 0.242·17-s + 1.14·19-s + 0.218·21-s − 0.208·23-s − 4/5·25-s + 0.192·27-s + 1.11·29-s − 1.07·31-s + 0.870·33-s − 0.169·35-s − 0.657·37-s + 0.800·39-s + 1.09·41-s + 1.06·43-s − 0.149·45-s + 0.875·47-s + 1/7·49-s + 0.140·51-s − 0.824·53-s − 0.674·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22848 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.933810803\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.933810803\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−15.63750108429212, −14.87793771777711, −14.24530545602002, −13.97048793461462, −13.62148386418932, −12.65174845808692, −12.24078519710248, −11.69537420720468, −11.09619763038043, −10.74355336122248, −9.774871255921635, −9.338981389014914, −8.814101068415306, −8.310405871022349, −7.587566011841973, −7.264443952672789, −6.275491630751878, −6.010379775544353, −5.041714373569031, −4.283605513654096, −3.652733260020100, −3.408526372910776, −2.284575080986233, −1.418361312704282, −0.8775937588071930,
0.8775937588071930, 1.418361312704282, 2.284575080986233, 3.408526372910776, 3.652733260020100, 4.283605513654096, 5.041714373569031, 6.010379775544353, 6.275491630751878, 7.264443952672789, 7.587566011841973, 8.310405871022349, 8.814101068415306, 9.338981389014914, 9.774871255921635, 10.74355336122248, 11.09619763038043, 11.69537420720468, 12.24078519710248, 12.65174845808692, 13.62148386418932, 13.97048793461462, 14.24530545602002, 14.87793771777711, 15.63750108429212
