| L(s) = 1 | + 3-s − 2·4-s + 3·5-s − 7-s − 2·9-s − 2·12-s + 4·13-s + 3·15-s + 4·16-s + 6·17-s − 2·19-s − 6·20-s − 21-s + 3·23-s + 4·25-s − 5·27-s + 2·28-s + 6·29-s + 5·31-s − 3·35-s + 4·36-s + 11·37-s + 4·39-s − 6·41-s − 8·43-s − 6·45-s + 4·48-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s + 1.34·5-s − 0.377·7-s − 2/3·9-s − 0.577·12-s + 1.10·13-s + 0.774·15-s + 16-s + 1.45·17-s − 0.458·19-s − 1.34·20-s − 0.218·21-s + 0.625·23-s + 4/5·25-s − 0.962·27-s + 0.377·28-s + 1.11·29-s + 0.898·31-s − 0.507·35-s + 2/3·36-s + 1.80·37-s + 0.640·39-s − 0.937·41-s − 1.21·43-s − 0.894·45-s + 0.577·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.872878656\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.872878656\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−9.833984807551462085646167334269, −9.487459835858535121288790482109, −8.508833833811404945218630536704, −8.085327774959512351361821751845, −6.46084267575588440942544784885, −5.81865396854508245636334597208, −4.95929836259470425310107953583, −3.61592205748787980286928494613, −2.75685822824512438972431635156, −1.19698223664384976616389110079,
1.19698223664384976616389110079, 2.75685822824512438972431635156, 3.61592205748787980286928494613, 4.95929836259470425310107953583, 5.81865396854508245636334597208, 6.46084267575588440942544784885, 8.085327774959512351361821751845, 8.508833833811404945218630536704, 9.487459835858535121288790482109, 9.833984807551462085646167334269
