| L(s) = 1 | + 2-s − 4-s + 2·7-s − 3·8-s − 4·13-s + 2·14-s − 16-s + 2·17-s + 2·23-s − 4·26-s − 2·28-s − 29-s + 4·31-s + 5·32-s + 2·34-s − 2·37-s − 10·41-s + 2·46-s + 12·47-s − 3·49-s + 4·52-s − 12·53-s − 6·56-s − 58-s − 4·59-s + 2·61-s + 4·62-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.755·7-s − 1.06·8-s − 1.10·13-s + 0.534·14-s − 1/4·16-s + 0.485·17-s + 0.417·23-s − 0.784·26-s − 0.377·28-s − 0.185·29-s + 0.718·31-s + 0.883·32-s + 0.342·34-s − 0.328·37-s − 1.56·41-s + 0.294·46-s + 1.75·47-s − 3/7·49-s + 0.554·52-s − 1.64·53-s − 0.801·56-s − 0.131·58-s − 0.520·59-s + 0.256·61-s + 0.508·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−7.68168901341573134752630452107, −6.91406710348089981133048404651, −6.06551652843038315302024645821, −5.23965904440638225697801207316, −4.87690647830632330784807770175, −4.16976943318245667324942020031, −3.28066046300377615286198001352, −2.51627531270622368055881277369, −1.35107218527809761708697311324, 0,
1.35107218527809761708697311324, 2.51627531270622368055881277369, 3.28066046300377615286198001352, 4.16976943318245667324942020031, 4.87690647830632330784807770175, 5.23965904440638225697801207316, 6.06551652843038315302024645821, 6.91406710348089981133048404651, 7.68168901341573134752630452107
