| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s − 2·7-s + 8-s + 9-s + 10-s − 3·11-s + 12-s − 13-s − 2·14-s + 15-s + 16-s + 18-s + 7·19-s + 20-s − 2·21-s − 3·22-s + 23-s + 24-s + 25-s − 26-s + 27-s − 2·28-s + 29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.904·11-s + 0.288·12-s − 0.277·13-s − 0.534·14-s + 0.258·15-s + 1/4·16-s + 0.235·18-s + 1.60·19-s + 0.223·20-s − 0.436·21-s − 0.639·22-s + 0.208·23-s + 0.204·24-s + 1/5·25-s − 0.196·26-s + 0.192·27-s − 0.377·28-s + 0.185·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 - 9 T + p T^{2} \) |
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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
−13.81080097244476, −13.45100319802737, −12.96938499773985, −12.58058748147503, −12.12649141048841, −11.54094621709070, −10.91213132978815, −10.45690506190767, −9.908402286294865, −9.539167287196788, −9.095986750162742, −8.327029837151793, −7.914494592212952, −7.249155458586290, −6.882068569562667, −6.427516040950577, −5.511477162521078, −5.290261221183898, −4.897560451814415, −3.878447902841229, −3.467628064767964, −3.037103383812380, −2.393737114268289, −1.861025038909208, −1.024231009291884, 0,
1.024231009291884, 1.861025038909208, 2.393737114268289, 3.037103383812380, 3.467628064767964, 3.878447902841229, 4.897560451814415, 5.290261221183898, 5.511477162521078, 6.427516040950577, 6.882068569562667, 7.249155458586290, 7.914494592212952, 8.327029837151793, 9.095986750162742, 9.539167287196788, 9.908402286294865, 10.45690506190767, 10.91213132978815, 11.54094621709070, 12.12649141048841, 12.58058748147503, 12.96938499773985, 13.45100319802737, 13.81080097244476
