| L(s) = 1 | − 3·3-s + 7.41·5-s + 9·9-s − 10.4·11-s + 2.78·13-s − 22.2·15-s + 50.4·17-s − 125.·19-s + 182.·23-s − 70.0·25-s − 27·27-s + 156.·29-s − 139.·31-s + 31.4·33-s − 394.·37-s − 8.36·39-s + 197.·41-s − 343.·43-s + 66.7·45-s + 610.·47-s − 151.·51-s − 137.·53-s − 77.7·55-s + 375.·57-s − 589.·59-s + 247.·61-s + 20.6·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.663·5-s + 0.333·9-s − 0.287·11-s + 0.0594·13-s − 0.382·15-s + 0.719·17-s − 1.50·19-s + 1.65·23-s − 0.560·25-s − 0.192·27-s + 0.999·29-s − 0.808·31-s + 0.165·33-s − 1.75·37-s − 0.0343·39-s + 0.752·41-s − 1.21·43-s + 0.221·45-s + 1.89·47-s − 0.415·51-s − 0.356·53-s − 0.190·55-s + 0.871·57-s − 1.30·59-s + 0.518·61-s + 0.0394·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 3T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 7.41T + 125T^{2} \) |
| 11 | \( 1 + 10.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 2.78T + 2.19e3T^{2} \) |
| 17 | \( 1 - 50.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 125.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 182.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 156.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 139.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 394.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 197.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 343.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 610.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 137.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 589.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 247.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 395.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 285.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 997.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 848.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 210.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 553.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 903.T + 9.12e5T^{2} \) |
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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
−8.341940118196285385880966887961, −7.30493726646146571687717749681, −6.65526351541382943907844805982, −5.83396653981419120732635513370, −5.21835744247113767680056530318, −4.36573635520243363203931192414, −3.27247053038598835852342152156, −2.19846270787411841900353541804, −1.22079080493526719760494007877, 0,
1.22079080493526719760494007877, 2.19846270787411841900353541804, 3.27247053038598835852342152156, 4.36573635520243363203931192414, 5.21835744247113767680056530318, 5.83396653981419120732635513370, 6.65526351541382943907844805982, 7.30493726646146571687717749681, 8.341940118196285385880966887961
