| L(s) = 1 | − 3·3-s − 14.7·5-s − 9.44·7-s + 9·9-s + 55.6·11-s + 13·13-s + 44.2·15-s − 120.·17-s − 136.·19-s + 28.3·21-s + 62.1·23-s + 92.4·25-s − 27·27-s + 171.·29-s + 68.3·31-s − 166.·33-s + 139.·35-s + 165.·37-s − 39·39-s − 437.·41-s + 191.·43-s − 132.·45-s + 271.·47-s − 253.·49-s + 360.·51-s + 155.·53-s − 820.·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.31·5-s − 0.510·7-s + 0.333·9-s + 1.52·11-s + 0.277·13-s + 0.761·15-s − 1.71·17-s − 1.65·19-s + 0.294·21-s + 0.563·23-s + 0.739·25-s − 0.192·27-s + 1.09·29-s + 0.396·31-s − 0.880·33-s + 0.672·35-s + 0.737·37-s − 0.160·39-s − 1.66·41-s + 0.678·43-s − 0.439·45-s + 0.842·47-s − 0.739·49-s + 0.990·51-s + 0.403·53-s − 2.01·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 3T \) |
| 13 | \( 1 - 13T \) |
| good | 5 | \( 1 + 14.7T + 125T^{2} \) |
| 7 | \( 1 + 9.44T + 343T^{2} \) |
| 11 | \( 1 - 55.6T + 1.33e3T^{2} \) |
| 17 | \( 1 + 120.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 136.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 62.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 171.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 68.3T + 2.97e4T^{2} \) |
| 37 | \( 1 - 165.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 437.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 191.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 271.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 155.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 582.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 261.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 197.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 69.8T + 3.57e5T^{2} \) |
| 73 | \( 1 - 815.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 396.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 938.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 572.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 145.T + 9.12e5T^{2} \) |
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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
−8.508694625259472564238117454684, −7.15718622969195994529227285240, −6.63780033222681765680477820983, −6.18790491381383220249275416522, −4.72862359920883960469774626487, −4.20092679535156054985683156844, −3.58268362094646846654012327315, −2.26694830278313360126549922682, −0.910113561309866894799434489924, 0,
0.910113561309866894799434489924, 2.26694830278313360126549922682, 3.58268362094646846654012327315, 4.20092679535156054985683156844, 4.72862359920883960469774626487, 6.18790491381383220249275416522, 6.63780033222681765680477820983, 7.15718622969195994529227285240, 8.508694625259472564238117454684
