L(s) = 1 | + 2-s + 2·3-s − 4-s + 5-s + 2·6-s − 2·7-s − 3·8-s + 9-s + 10-s − 2·12-s + 13-s − 2·14-s + 2·15-s − 16-s − 5·17-s + 18-s + 6·19-s − 20-s − 4·21-s + 2·23-s − 6·24-s − 4·25-s + 26-s − 4·27-s + 2·28-s + 9·29-s + 2·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.15·3-s − 1/2·4-s + 0.447·5-s + 0.816·6-s − 0.755·7-s − 1.06·8-s + 1/3·9-s + 0.316·10-s − 0.577·12-s + 0.277·13-s − 0.534·14-s + 0.516·15-s − 1/4·16-s − 1.21·17-s + 0.235·18-s + 1.37·19-s − 0.223·20-s − 0.872·21-s + 0.417·23-s − 1.22·24-s − 4/5·25-s + 0.196·26-s − 0.769·27-s + 0.377·28-s + 1.67·29-s + 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.666156920\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.666156920\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−13.65039157651840903490685790066, −12.93331549622800533527021314779, −11.69772641079315514179053119720, −9.975450561586109317601560382100, −9.182457967238804819066801847611, −8.356936232923193217562617522792, −6.72221402891887535126565651102, −5.38415572352897256696127903868, −3.84177902362047353880486406829, −2.75274084463573300677827364197,
2.75274084463573300677827364197, 3.84177902362047353880486406829, 5.38415572352897256696127903868, 6.72221402891887535126565651102, 8.356936232923193217562617522792, 9.182457967238804819066801847611, 9.975450561586109317601560382100, 11.69772641079315514179053119720, 12.93331549622800533527021314779, 13.65039157651840903490685790066