| L(s) = 1 | − 2-s − 4-s + 3·8-s + 11-s − 4·13-s − 16-s − 4·17-s − 8·19-s − 22-s + 8·23-s − 5·25-s + 4·26-s − 2·29-s − 4·31-s − 5·32-s + 4·34-s + 10·37-s + 8·38-s − 4·41-s + 4·43-s − 44-s − 8·46-s + 12·47-s + 5·50-s + 4·52-s − 6·53-s + 2·58-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 1.06·8-s + 0.301·11-s − 1.10·13-s − 1/4·16-s − 0.970·17-s − 1.83·19-s − 0.213·22-s + 1.66·23-s − 25-s + 0.784·26-s − 0.371·29-s − 0.718·31-s − 0.883·32-s + 0.685·34-s + 1.64·37-s + 1.29·38-s − 0.624·41-s + 0.609·43-s − 0.150·44-s − 1.17·46-s + 1.75·47-s + 0.707·50-s + 0.554·52-s − 0.824·53-s + 0.262·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4851 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4851 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6694244679\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6694244679\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−8.409385416542134183695339119891, −7.60221778266205152274240531932, −7.05628385181258628112854758836, −6.21557954341212684879077517232, −5.25625833674742948887238385228, −4.45349867227065303447375727337, −4.00566710697544403731985696361, −2.63124203188893512308933636446, −1.81597492238485084860043398929, −0.49537148228846709068364932739,
0.49537148228846709068364932739, 1.81597492238485084860043398929, 2.63124203188893512308933636446, 4.00566710697544403731985696361, 4.45349867227065303447375727337, 5.25625833674742948887238385228, 6.21557954341212684879077517232, 7.05628385181258628112854758836, 7.60221778266205152274240531932, 8.409385416542134183695339119891
