| L(s) = 1 | − 3·3-s + 11·5-s − 19·7-s − 18·9-s + 38·11-s − 13·13-s − 33·15-s − 51·17-s − 90·19-s + 57·21-s + 52·23-s − 4·25-s + 135·27-s − 190·29-s − 292·31-s − 114·33-s − 209·35-s − 441·37-s + 39·39-s + 312·41-s − 373·43-s − 198·45-s + 41·47-s + 18·49-s + 153·51-s + 468·53-s + 418·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.983·5-s − 1.02·7-s − 2/3·9-s + 1.04·11-s − 0.277·13-s − 0.568·15-s − 0.727·17-s − 1.08·19-s + 0.592·21-s + 0.471·23-s − 0.0319·25-s + 0.962·27-s − 1.21·29-s − 1.69·31-s − 0.601·33-s − 1.00·35-s − 1.95·37-s + 0.160·39-s + 1.18·41-s − 1.32·43-s − 0.655·45-s + 0.127·47-s + 0.0524·49-s + 0.420·51-s + 1.21·53-s + 1.02·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + p T \) |
| good | 3 | \( 1 + p T + p^{3} T^{2} \) |
| 5 | \( 1 - 11 T + p^{3} T^{2} \) |
| 7 | \( 1 + 19 T + p^{3} T^{2} \) |
| 11 | \( 1 - 38 T + p^{3} T^{2} \) |
| 17 | \( 1 + 3 p T + p^{3} T^{2} \) |
| 19 | \( 1 + 90 T + p^{3} T^{2} \) |
| 23 | \( 1 - 52 T + p^{3} T^{2} \) |
| 29 | \( 1 + 190 T + p^{3} T^{2} \) |
| 31 | \( 1 + 292 T + p^{3} T^{2} \) |
| 37 | \( 1 + 441 T + p^{3} T^{2} \) |
| 41 | \( 1 - 312 T + p^{3} T^{2} \) |
| 43 | \( 1 + 373 T + p^{3} T^{2} \) |
| 47 | \( 1 - 41 T + p^{3} T^{2} \) |
| 53 | \( 1 - 468 T + p^{3} T^{2} \) |
| 59 | \( 1 + 530 T + p^{3} T^{2} \) |
| 61 | \( 1 - 592 T + p^{3} T^{2} \) |
| 67 | \( 1 - 206 T + p^{3} T^{2} \) |
| 71 | \( 1 - 863 T + p^{3} T^{2} \) |
| 73 | \( 1 + 322 T + p^{3} T^{2} \) |
| 79 | \( 1 - 460 T + p^{3} T^{2} \) |
| 83 | \( 1 + 528 T + p^{3} T^{2} \) |
| 89 | \( 1 - 870 T + p^{3} T^{2} \) |
| 97 | \( 1 + 346 T + p^{3} T^{2} \) |
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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
−11.39159440435699193192974427217, −10.51302449925206438629812467387, −9.415071880037936337375208451838, −8.815381300303618973458573123727, −6.93539422371268789491404258253, −6.23021524907193253317949272978, −5.32353669737643752737942217869, −3.66718828268372373545706421377, −2.05394510222426434168592675448, 0,
2.05394510222426434168592675448, 3.66718828268372373545706421377, 5.32353669737643752737942217869, 6.23021524907193253317949272978, 6.93539422371268789491404258253, 8.815381300303618973458573123727, 9.415071880037936337375208451838, 10.51302449925206438629812467387, 11.39159440435699193192974427217
