| L(s) = 1 | + 3.16·3-s − 5·5-s − 10.1·7-s − 17·9-s + 21.9·11-s − 63.6·13-s − 15.8·15-s − 119.·17-s − 116.·19-s − 32.1·21-s + 206.·23-s + 25·25-s − 139.·27-s + 160.·29-s + 158.·31-s + 69.4·33-s + 50.7·35-s + 225.·37-s − 201.·39-s + 269.·41-s + 469.·43-s + 85·45-s − 21.4·47-s − 239.·49-s − 377.·51-s − 269.·53-s − 109.·55-s + ⋯ |
| L(s) = 1 | + 0.608·3-s − 0.447·5-s − 0.548·7-s − 0.629·9-s + 0.601·11-s − 1.35·13-s − 0.272·15-s − 1.70·17-s − 1.41·19-s − 0.333·21-s + 1.87·23-s + 0.200·25-s − 0.991·27-s + 1.03·29-s + 0.916·31-s + 0.366·33-s + 0.245·35-s + 1.00·37-s − 0.826·39-s + 1.02·41-s + 1.66·43-s + 0.281·45-s − 0.0666·47-s − 0.699·49-s − 1.03·51-s − 0.699·53-s − 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.470782610\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.470782610\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| good | 3 | \( 1 - 3.16T + 27T^{2} \) |
| 7 | \( 1 + 10.1T + 343T^{2} \) |
| 11 | \( 1 - 21.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 63.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 119.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 116.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 206.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 160.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 158.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 225.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 269.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 469.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 21.4T + 1.03e5T^{2} \) |
| 53 | \( 1 + 269.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 556.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 242.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 769.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 550.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 136.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 43.2T + 4.93e5T^{2} \) |
| 83 | \( 1 + 113.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 172.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 558.T + 9.12e5T^{2} \) |
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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
−9.038569785460256845168530439333, −8.728629289604967914349267749550, −7.70826038959838318825339691723, −6.82132990740846692874699399455, −6.21251262333779085486483648497, −4.79141983230172780174929875854, −4.17722877227563074753298576554, −2.88047579252972107433521658025, −2.37372391857771756332400732776, −0.56358620997226360651167385550,
0.56358620997226360651167385550, 2.37372391857771756332400732776, 2.88047579252972107433521658025, 4.17722877227563074753298576554, 4.79141983230172780174929875854, 6.21251262333779085486483648497, 6.82132990740846692874699399455, 7.70826038959838318825339691723, 8.728629289604967914349267749550, 9.038569785460256845168530439333
