| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s + 4·7-s − 8-s + 9-s − 10-s + 3·11-s − 12-s + 13-s − 4·14-s − 15-s + 16-s + 2·17-s − 18-s − 7·19-s + 20-s − 4·21-s − 3·22-s + 7·23-s + 24-s + 25-s − 26-s − 27-s + 4·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 1.51·7-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 − 1.06·14-s − 0.258·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s − 1.60·19-s + 0.223·20-s − 0.872·21-s − 0.639·22-s + 1.45·23-s + 0.204·24-s + 1/5·25-s − 0.196·26-s − 0.192·27-s + 0.755·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 374790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 374790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.273582648\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.273582648\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 31 | \( 1 \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−12.29784066304721, −11.92397225623212, −11.48478993938485, −11.06810635624395, −10.72987073205363, −10.29101405916356, −9.933295093962768, −9.111642256371520, −8.864188753683120, −8.532773792878915, −7.937415797354789, −7.518437951589778, −6.943050909078404, −6.456458377567967, −6.133657921265932, −5.595896764564161, −4.889079238692373, −4.608559510559882, −4.184492521663898, −3.309893487306267, −2.777869158393502, −1.943039189812899, −1.690585579273201, −1.006356890668092, −0.6438202481644123,
0.6438202481644123, 1.006356890668092, 1.690585579273201, 1.943039189812899, 2.777869158393502, 3.309893487306267, 4.184492521663898, 4.608559510559882, 4.889079238692373, 5.595896764564161, 6.133657921265932, 6.456458377567967, 6.943050909078404, 7.518437951589778, 7.937415797354789, 8.532773792878915, 8.864188753683120, 9.111642256371520, 9.933295093962768, 10.29101405916356, 10.72987073205363, 11.06810635624395, 11.48478993938485, 11.92397225623212, 12.29784066304721
