| L(s) = 1 | − 2-s + 3-s + 4-s − 3·5-s − 6-s − 8-s + 9-s + 3·10-s + 12-s − 13-s − 3·15-s + 16-s + 17-s − 18-s − 4·19-s − 3·20-s + 6·23-s − 24-s + 4·25-s + 26-s + 27-s + 3·29-s + 3·30-s − 31-s − 32-s − 34-s + 36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.34·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.948·10-s + 0.288·12-s − 0.277·13-s − 0.774·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s − 0.917·19-s − 0.670·20-s + 1.25·23-s − 0.204·24-s + 4/5·25-s + 0.196·26-s + 0.192·27-s + 0.557·29-s + 0.547·30-s − 0.179·31-s − 0.176·32-s − 0.171·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4998 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4998 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 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
−7.970004144622494835745038586731, −7.23984792999900623002141226729, −6.93453067343474438716682948881, −5.81064900227063250971473122831, −4.74123851218510371439634228966, −4.02590477555712259831328038478, −3.22048183814887169917135483913, −2.44512797639457379031527980047, −1.20581900863246518921637570841, 0,
1.20581900863246518921637570841, 2.44512797639457379031527980047, 3.22048183814887169917135483913, 4.02590477555712259831328038478, 4.74123851218510371439634228966, 5.81064900227063250971473122831, 6.93453067343474438716682948881, 7.23984792999900623002141226729, 7.970004144622494835745038586731
