| L(s) = 1 | − 2-s + 3-s + 4-s − 2·5-s − 6-s − 8-s + 9-s + 2·10-s − 3·11-s + 12-s − 4·13-s − 2·15-s + 16-s − 6·17-s − 18-s − 2·20-s + 3·22-s − 24-s − 25-s + 4·26-s + 27-s + 3·29-s + 2·30-s + 31-s − 32-s − 3·33-s + 6·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.632·10-s − 0.904·11-s + 0.288·12-s − 1.10·13-s − 0.516·15-s + 1/4·16-s − 1.45·17-s − 0.235·18-s − 0.447·20-s + 0.639·22-s − 0.204·24-s − 1/5·25-s + 0.784·26-s + 0.192·27-s + 0.557·29-s + 0.365·30-s + 0.179·31-s − 0.176·32-s − 0.522·33-s + 1.02·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 155526 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155526 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.362968395\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.362968395\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 12 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 13 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 + 9 T + p T^{2} \) |
| 83 | \( 1 - 5 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 15 T + p T^{2} \) |
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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
−13.28810303555299, −12.81474268469473, −12.31118722779483, −11.80340352809314, −11.38556790495819, −10.82454498349432, −10.46471742594881, −9.877507338615759, −9.404116602818495, −8.978397950988274, −8.395732825684806, −7.996205323324940, −7.575288558632691, −7.158327504616235, −6.737816443185396, −5.920039984648639, −5.465273183661528, −4.502350445939794, −4.382106671134788, −3.723207389997482, −2.838688973656743, −2.372214069835976, −2.226257446417414, −0.9217672051327226, −0.4507177249174147,
0.4507177249174147, 0.9217672051327226, 2.226257446417414, 2.372214069835976, 2.838688973656743, 3.723207389997482, 4.382106671134788, 4.502350445939794, 5.465273183661528, 5.920039984648639, 6.737816443185396, 7.158327504616235, 7.575288558632691, 7.996205323324940, 8.395732825684806, 8.978397950988274, 9.404116602818495, 9.877507338615759, 10.46471742594881, 10.82454498349432, 11.38556790495819, 11.80340352809314, 12.31118722779483, 12.81474268469473, 13.28810303555299
