L(s) = 1 | + 0.115·2-s − 2.98·3-s − 1.98·4-s − 0.455·5-s − 0.345·6-s − 0.462·8-s + 5.89·9-s − 0.0528·10-s + 3.02·11-s + 5.92·12-s + 0.114·13-s + 1.35·15-s + 3.91·16-s + 0.380·17-s + 0.683·18-s − 19-s + 0.904·20-s + 0.350·22-s + 7.55·23-s + 1.37·24-s − 4.79·25-s + 0.0132·26-s − 8.61·27-s − 7.78·29-s + 0.157·30-s − 6.78·31-s + 1.37·32-s + ⋯ |
L(s) = 1 | + 0.0820·2-s − 1.72·3-s − 0.993·4-s − 0.203·5-s − 0.141·6-s − 0.163·8-s + 1.96·9-s − 0.0167·10-s + 0.911·11-s + 1.70·12-s + 0.0316·13-s + 0.350·15-s + 0.979·16-s + 0.0921·17-s + 0.161·18-s − 0.229·19-s + 0.202·20-s + 0.0747·22-s + 1.57·23-s + 0.281·24-s − 0.958·25-s + 0.00259·26-s − 1.65·27-s − 1.44·29-s + 0.0287·30-s − 1.21·31-s + 0.243·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 2 | \( 1 - 0.115T + 2T^{2} \) |
| 3 | \( 1 + 2.98T + 3T^{2} \) |
| 5 | \( 1 + 0.455T + 5T^{2} \) |
| 11 | \( 1 - 3.02T + 11T^{2} \) |
| 13 | \( 1 - 0.114T + 13T^{2} \) |
| 17 | \( 1 - 0.380T + 17T^{2} \) |
| 23 | \( 1 - 7.55T + 23T^{2} \) |
| 29 | \( 1 + 7.78T + 29T^{2} \) |
| 31 | \( 1 + 6.78T + 31T^{2} \) |
| 37 | \( 1 + 1.55T + 37T^{2} \) |
| 41 | \( 1 - 6.21T + 41T^{2} \) |
| 43 | \( 1 - 9.25T + 43T^{2} \) |
| 47 | \( 1 + 11.1T + 47T^{2} \) |
| 53 | \( 1 + 8.04T + 53T^{2} \) |
| 59 | \( 1 - 7.35T + 59T^{2} \) |
| 61 | \( 1 + 12.5T + 61T^{2} \) |
| 67 | \( 1 + 2.62T + 67T^{2} \) |
| 71 | \( 1 - 3.18T + 71T^{2} \) |
| 73 | \( 1 - 9.01T + 73T^{2} \) |
| 79 | \( 1 + 10.8T + 79T^{2} \) |
| 83 | \( 1 - 3.10T + 83T^{2} \) |
| 89 | \( 1 + 4.12T + 89T^{2} \) |
| 97 | \( 1 - 15.6T + 97T^{2} \) |
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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
−9.554971790361386462718659656404, −9.165459701562662753514903364835, −7.81201915772127649873105568476, −6.91444927042994828329576747938, −5.96081733814866279498058229310, −5.32029180463887611097034603603, −4.44917347117438314223470679550, −3.65187988550218330604301043548, −1.32929827621645345330969015361, 0,
1.32929827621645345330969015361, 3.65187988550218330604301043548, 4.44917347117438314223470679550, 5.32029180463887611097034603603, 5.96081733814866279498058229310, 6.91444927042994828329576747938, 7.81201915772127649873105568476, 9.165459701562662753514903364835, 9.554971790361386462718659656404