| L(s) = 1 | + 2·2-s + 3·3-s − 4·4-s + 6·6-s − 29·7-s − 24·8-s + 9·9-s − 15·11-s − 12·12-s − 3·13-s − 58·14-s − 16·16-s − 121·17-s + 18·18-s − 40·19-s − 87·21-s − 30·22-s + 116·23-s − 72·24-s − 6·26-s + 27·27-s + 116·28-s + 29·29-s − 116·31-s + 160·32-s − 45·33-s − 242·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.408·6-s − 1.56·7-s − 1.06·8-s + 1/3·9-s − 0.411·11-s − 0.288·12-s − 0.0640·13-s − 1.10·14-s − 1/4·16-s − 1.72·17-s + 0.235·18-s − 0.482·19-s − 0.904·21-s − 0.290·22-s + 1.05·23-s − 0.612·24-s − 0.0452·26-s + 0.192·27-s + 0.782·28-s + 0.185·29-s − 0.672·31-s + 0.883·32-s − 0.237·33-s − 1.22·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.189043231\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.189043231\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - p T \) |
| 5 | \( 1 \) |
| 29 | \( 1 - p T \) |
| good | 2 | \( 1 - p T + p^{3} T^{2} \) |
| 7 | \( 1 + 29 T + p^{3} T^{2} \) |
| 11 | \( 1 + 15 T + p^{3} T^{2} \) |
| 13 | \( 1 + 3 T + p^{3} T^{2} \) |
| 17 | \( 1 + 121 T + p^{3} T^{2} \) |
| 19 | \( 1 + 40 T + p^{3} T^{2} \) |
| 23 | \( 1 - 116 T + p^{3} T^{2} \) |
| 31 | \( 1 + 116 T + p^{3} T^{2} \) |
| 37 | \( 1 + 36 T + p^{3} T^{2} \) |
| 41 | \( 1 + 170 T + p^{3} T^{2} \) |
| 43 | \( 1 + 230 T + p^{3} T^{2} \) |
| 47 | \( 1 + 231 T + p^{3} T^{2} \) |
| 53 | \( 1 + 456 T + p^{3} T^{2} \) |
| 59 | \( 1 - 576 T + p^{3} T^{2} \) |
| 61 | \( 1 - 342 T + p^{3} T^{2} \) |
| 67 | \( 1 - 269 T + p^{3} T^{2} \) |
| 71 | \( 1 - 302 T + p^{3} T^{2} \) |
| 73 | \( 1 - 372 T + p^{3} T^{2} \) |
| 79 | \( 1 + 348 T + p^{3} T^{2} \) |
| 83 | \( 1 - 512 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1525 T + p^{3} T^{2} \) |
| 97 | \( 1 - 560 T + p^{3} T^{2} \) |
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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
−8.901771229429447828472880725787, −8.086693052225951071006358066087, −6.70660032870881826669640995342, −6.61147638131663082510602025789, −5.39255163315108938734744629868, −4.60988958650577999063766697157, −3.70469960008474341485642605084, −3.10009871281821382929517719633, −2.21706332085147978873045090542, −0.40363430113719401569140431965,
0.40363430113719401569140431965, 2.21706332085147978873045090542, 3.10009871281821382929517719633, 3.70469960008474341485642605084, 4.60988958650577999063766697157, 5.39255163315108938734744629868, 6.61147638131663082510602025789, 6.70660032870881826669640995342, 8.086693052225951071006358066087, 8.901771229429447828472880725787
