| L(s) = 1 | − 3-s − 2·5-s + 9-s − 13-s + 2·15-s + 2·17-s + 4·19-s − 25-s − 27-s − 6·29-s + 2·37-s + 39-s + 6·41-s + 12·43-s − 2·45-s − 4·47-s − 7·49-s − 2·51-s − 6·53-s − 4·57-s + 8·59-s + 2·61-s + 2·65-s − 4·67-s − 12·71-s − 14·73-s + 75-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 1/3·9-s − 0.277·13-s + 0.516·15-s + 0.485·17-s + 0.917·19-s − 1/5·25-s − 0.192·27-s − 1.11·29-s + 0.328·37-s + 0.160·39-s + 0.937·41-s + 1.82·43-s − 0.298·45-s − 0.583·47-s − 49-s − 0.280·51-s − 0.824·53-s − 0.529·57-s + 1.04·59-s + 0.256·61-s + 0.248·65-s − 0.488·67-s − 1.42·71-s − 1.63·73-s + 0.115·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−8.429645114873483352252590672860, −7.51727934813324433491134862469, −7.30784619649759594552263736405, −6.08537646602212143968327685330, −5.46962489281230822667740001129, −4.49909620784530543421734041489, −3.79070284272267126319936744658, −2.78518661391770994956776280001, −1.32676341715050635172211285974, 0,
1.32676341715050635172211285974, 2.78518661391770994956776280001, 3.79070284272267126319936744658, 4.49909620784530543421734041489, 5.46962489281230822667740001129, 6.08537646602212143968327685330, 7.30784619649759594552263736405, 7.51727934813324433491134862469, 8.429645114873483352252590672860
