| L(s) = 1 | + 3·3-s − 8·5-s − 12·7-s + 9·9-s − 12·11-s + 20·13-s − 24·15-s + 62·17-s + 108·19-s − 36·21-s + 72·23-s − 61·25-s + 27·27-s − 128·29-s − 204·31-s − 36·33-s + 96·35-s − 228·37-s + 60·39-s + 22·41-s − 204·43-s − 72·45-s − 600·47-s − 199·49-s + 186·51-s + 256·53-s + 96·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.715·5-s − 0.647·7-s + 1/3·9-s − 0.328·11-s + 0.426·13-s − 0.413·15-s + 0.884·17-s + 1.30·19-s − 0.374·21-s + 0.652·23-s − 0.487·25-s + 0.192·27-s − 0.819·29-s − 1.18·31-s − 0.189·33-s + 0.463·35-s − 1.01·37-s + 0.246·39-s + 0.0838·41-s − 0.723·43-s − 0.238·45-s − 1.86·47-s − 0.580·49-s + 0.510·51-s + 0.663·53-s + 0.235·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - p T \) |
| good | 5 | \( 1 + 8 T + p^{3} T^{2} \) |
| 7 | \( 1 + 12 T + p^{3} T^{2} \) |
| 11 | \( 1 + 12 T + p^{3} T^{2} \) |
| 13 | \( 1 - 20 T + p^{3} T^{2} \) |
| 17 | \( 1 - 62 T + p^{3} T^{2} \) |
| 19 | \( 1 - 108 T + p^{3} T^{2} \) |
| 23 | \( 1 - 72 T + p^{3} T^{2} \) |
| 29 | \( 1 + 128 T + p^{3} T^{2} \) |
| 31 | \( 1 + 204 T + p^{3} T^{2} \) |
| 37 | \( 1 + 228 T + p^{3} T^{2} \) |
| 41 | \( 1 - 22 T + p^{3} T^{2} \) |
| 43 | \( 1 + 204 T + p^{3} T^{2} \) |
| 47 | \( 1 + 600 T + p^{3} T^{2} \) |
| 53 | \( 1 - 256 T + p^{3} T^{2} \) |
| 59 | \( 1 + 828 T + p^{3} T^{2} \) |
| 61 | \( 1 + 84 T + p^{3} T^{2} \) |
| 67 | \( 1 - 348 T + p^{3} T^{2} \) |
| 71 | \( 1 + 456 T + p^{3} T^{2} \) |
| 73 | \( 1 + 822 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1356 T + p^{3} T^{2} \) |
| 83 | \( 1 - 108 T + p^{3} T^{2} \) |
| 89 | \( 1 - 938 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1278 T + p^{3} T^{2} \) |
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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
−9.491163897258089000615740886523, −8.648244064526370482747213349130, −7.68415481686808920765895431061, −7.19788859416028818586791013968, −5.94501137584810084902799689365, −4.93458510438755303957203816276, −3.55584382040405851196106792860, −3.19587389648295029777001640025, −1.54154597420497502404888668768, 0,
1.54154597420497502404888668768, 3.19587389648295029777001640025, 3.55584382040405851196106792860, 4.93458510438755303957203816276, 5.94501137584810084902799689365, 7.19788859416028818586791013968, 7.68415481686808920765895431061, 8.648244064526370482747213349130, 9.491163897258089000615740886523
