| L(s) = 1 | − 3.25e4·3-s + 5.76e6·5-s + 1.36e8·7-s − 1.03e8·9-s − 9.46e9·11-s − 2.45e10·13-s − 1.87e11·15-s − 6.57e11·17-s + 2.38e12·19-s − 4.42e12·21-s + 9.87e11·23-s + 1.41e13·25-s + 4.11e13·27-s + 3.10e13·29-s + 3.73e13·31-s + 3.07e14·33-s + 7.84e14·35-s − 1.48e15·37-s + 7.98e14·39-s + 2.50e14·41-s − 5.88e15·43-s − 5.98e14·45-s − 6.94e15·47-s + 7.13e15·49-s + 2.13e16·51-s + 1.66e16·53-s − 5.45e16·55-s + ⋯ |
| L(s) = 1 | − 0.954·3-s + 1.32·5-s + 1.27·7-s − 0.0893·9-s − 1.21·11-s − 0.641·13-s − 1.26·15-s − 1.34·17-s + 1.69·19-s − 1.21·21-s + 0.114·23-s + 0.743·25-s + 1.03·27-s + 0.397·29-s + 0.253·31-s + 1.15·33-s + 1.68·35-s − 1.88·37-s + 0.612·39-s + 0.119·41-s − 1.78·43-s − 0.117·45-s − 0.905·47-s + 0.625·49-s + 1.28·51-s + 0.693·53-s − 1.59·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(10)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{21}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + 3.25e4T + 1.16e9T^{2} \) |
| 5 | \( 1 - 5.76e6T + 1.90e13T^{2} \) |
| 7 | \( 1 - 1.36e8T + 1.13e16T^{2} \) |
| 11 | \( 1 + 9.46e9T + 6.11e19T^{2} \) |
| 13 | \( 1 + 2.45e10T + 1.46e21T^{2} \) |
| 17 | \( 1 + 6.57e11T + 2.39e23T^{2} \) |
| 19 | \( 1 - 2.38e12T + 1.97e24T^{2} \) |
| 23 | \( 1 - 9.87e11T + 7.46e25T^{2} \) |
| 29 | \( 1 - 3.10e13T + 6.10e27T^{2} \) |
| 31 | \( 1 - 3.73e13T + 2.16e28T^{2} \) |
| 37 | \( 1 + 1.48e15T + 6.24e29T^{2} \) |
| 41 | \( 1 - 2.50e14T + 4.39e30T^{2} \) |
| 43 | \( 1 + 5.88e15T + 1.08e31T^{2} \) |
| 47 | \( 1 + 6.94e15T + 5.88e31T^{2} \) |
| 53 | \( 1 - 1.66e16T + 5.77e32T^{2} \) |
| 59 | \( 1 - 4.93e16T + 4.42e33T^{2} \) |
| 61 | \( 1 + 4.46e16T + 8.34e33T^{2} \) |
| 67 | \( 1 - 2.92e17T + 4.95e34T^{2} \) |
| 71 | \( 1 - 3.11e17T + 1.49e35T^{2} \) |
| 73 | \( 1 - 2.24e17T + 2.53e35T^{2} \) |
| 79 | \( 1 + 1.57e18T + 1.13e36T^{2} \) |
| 83 | \( 1 - 4.67e17T + 2.90e36T^{2} \) |
| 89 | \( 1 + 5.81e18T + 1.09e37T^{2} \) |
| 97 | \( 1 + 5.35e18T + 5.60e37T^{2} \) |
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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
−11.88199129164130359182815800856, −10.90591760284986243832579620340, −9.893732731723763295101029485265, −8.361942253350585871741865313564, −6.82648072266379930280851193394, −5.29485446514471291254669111505, −5.09599786646884192619486263866, −2.58190181630710700272865188455, −1.47702575524254578765710734117, 0,
1.47702575524254578765710734117, 2.58190181630710700272865188455, 5.09599786646884192619486263866, 5.29485446514471291254669111505, 6.82648072266379930280851193394, 8.361942253350585871741865313564, 9.893732731723763295101029485265, 10.90591760284986243832579620340, 11.88199129164130359182815800856
