L(s) = 1 | − 3-s − 2·4-s − 7-s − 2·9-s − 3·11-s + 2·12-s − 5·13-s + 4·16-s − 3·17-s + 2·19-s + 21-s + 6·23-s + 5·27-s + 2·28-s + 3·29-s − 4·31-s + 3·33-s + 4·36-s − 2·37-s + 5·39-s − 12·41-s + 10·43-s + 6·44-s − 9·47-s − 4·48-s + 49-s + 3·51-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 4-s − 0.377·7-s − 2/3·9-s − 0.904·11-s + 0.577·12-s − 1.38·13-s + 16-s − 0.727·17-s + 0.458·19-s + 0.218·21-s + 1.25·23-s + 0.962·27-s + 0.377·28-s + 0.557·29-s − 0.718·31-s + 0.522·33-s + 2/3·36-s − 0.328·37-s + 0.800·39-s − 1.87·41-s + 1.52·43-s + 0.904·44-s − 1.31·47-s − 0.577·48-s + 1/7·49-s + 0.420·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 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
−12.38708666154083380388884256770, −11.22233622493928875067803778362, −10.17729908145776565325006315809, −9.261964485898248388362511716034, −8.214592059880155258547933248387, −6.93454011637102289378074371468, −5.45020178597402395728815947142, −4.78686073935120916368506592009, −2.98704088538274633622207409794, 0,
2.98704088538274633622207409794, 4.78686073935120916368506592009, 5.45020178597402395728815947142, 6.93454011637102289378074371468, 8.214592059880155258547933248387, 9.261964485898248388362511716034, 10.17729908145776565325006315809, 11.22233622493928875067803778362, 12.38708666154083380388884256770