| L(s) = 1 | + 0.660·2-s + 9.58·3-s − 7.56·4-s + 9.05·5-s + 6.32·6-s − 10.2·8-s + 64.8·9-s + 5.97·10-s − 55.8·11-s − 72.4·12-s + 12.5·13-s + 86.7·15-s + 53.7·16-s − 32.7·17-s + 42.7·18-s − 69.5·19-s − 68.4·20-s − 36.8·22-s − 26.4·23-s − 98.4·24-s − 43.0·25-s + 8.27·26-s + 362.·27-s + 31.4·29-s + 57.2·30-s − 195.·31-s + 117.·32-s + ⋯ |
| L(s) = 1 | + 0.233·2-s + 1.84·3-s − 0.945·4-s + 0.809·5-s + 0.430·6-s − 0.454·8-s + 2.40·9-s + 0.189·10-s − 1.53·11-s − 1.74·12-s + 0.267·13-s + 1.49·15-s + 0.839·16-s − 0.467·17-s + 0.560·18-s − 0.839·19-s − 0.765·20-s − 0.357·22-s − 0.239·23-s − 0.837·24-s − 0.344·25-s + 0.0624·26-s + 2.58·27-s + 0.201·29-s + 0.348·30-s − 1.13·31-s + 0.650·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| good | 2 | \( 1 - 0.660T + 8T^{2} \) |
| 3 | \( 1 - 9.58T + 27T^{2} \) |
| 5 | \( 1 - 9.05T + 125T^{2} \) |
| 11 | \( 1 + 55.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 12.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 32.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 69.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 26.4T + 1.21e4T^{2} \) |
| 29 | \( 1 - 31.4T + 2.43e4T^{2} \) |
| 31 | \( 1 + 195.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 362.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 350.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 207.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 366.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 227.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 532.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 20.7T + 2.26e5T^{2} \) |
| 67 | \( 1 + 682.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 297.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 308.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 804.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.37e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 524.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 251.T + 9.12e5T^{2} \) |
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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
−8.284229410791528435480111941169, −7.85114657953980765798986867422, −6.83601420029898467586898980704, −5.71438395896413055697687396628, −4.89917595849333380478310837143, −4.03992541023037046693414608803, −3.24963450527495453625114045526, −2.40689835257253774920446326531, −1.66699970295812780961945971078, 0,
1.66699970295812780961945971078, 2.40689835257253774920446326531, 3.24963450527495453625114045526, 4.03992541023037046693414608803, 4.89917595849333380478310837143, 5.71438395896413055697687396628, 6.83601420029898467586898980704, 7.85114657953980765798986867422, 8.284229410791528435480111941169
