| L(s) = 1 | − 2-s + 2·3-s + 4-s − 2·6-s − 7-s − 8-s + 9-s − 4·11-s + 2·12-s + 14-s + 16-s − 6·17-s − 18-s + 6·19-s − 2·21-s + 4·22-s − 2·24-s − 5·25-s − 4·27-s − 28-s + 10·29-s + 4·31-s − 32-s − 8·33-s + 6·34-s + 36-s + 2·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.816·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 1.20·11-s + 0.577·12-s + 0.267·14-s + 1/4·16-s − 1.45·17-s − 0.235·18-s + 1.37·19-s − 0.436·21-s + 0.852·22-s − 0.408·24-s − 25-s − 0.769·27-s − 0.188·28-s + 1.85·29-s + 0.718·31-s − 0.176·32-s − 1.39·33-s + 1.02·34-s + 1/6·36-s + 0.328·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7406 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7406 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.612673959\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.612673959\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−7.935750723724303395085728565043, −7.51762998310089004413694218668, −6.73512835485393867583478660346, −5.95032633647146751639425965714, −5.09031295537220141485050290587, −4.16723605842224367829088208544, −3.17369685166786963729368212534, −2.66134514657213565885767413123, −2.01519638434642891946945745227, −0.63972369986057906382968639971,
0.63972369986057906382968639971, 2.01519638434642891946945745227, 2.66134514657213565885767413123, 3.17369685166786963729368212534, 4.16723605842224367829088208544, 5.09031295537220141485050290587, 5.95032633647146751639425965714, 6.73512835485393867583478660346, 7.51762998310089004413694218668, 7.935750723724303395085728565043
