| L(s) = 1 | + 230.·3-s − 2.53e3·5-s + 2.40e3·7-s + 3.32e4·9-s − 5.30e3·11-s − 1.38e5·13-s − 5.82e5·15-s + 1.29e5·17-s − 8.86e5·19-s + 5.52e5·21-s − 1.67e6·23-s + 4.45e6·25-s + 3.11e6·27-s − 4.30e6·29-s + 2.81e6·31-s − 1.22e6·33-s − 6.08e6·35-s − 1.40e7·37-s − 3.17e7·39-s + 1.56e7·41-s − 4.41e7·43-s − 8.41e7·45-s + 4.79e7·47-s + 5.76e6·49-s + 2.98e7·51-s + 3.77e7·53-s + 1.34e7·55-s + ⋯ |
| L(s) = 1 | + 1.63·3-s − 1.81·5-s + 0.377·7-s + 1.68·9-s − 0.109·11-s − 1.34·13-s − 2.97·15-s + 0.376·17-s − 1.56·19-s + 0.619·21-s − 1.24·23-s + 2.28·25-s + 1.12·27-s − 1.13·29-s + 0.546·31-s − 0.179·33-s − 0.684·35-s − 1.23·37-s − 2.19·39-s + 0.866·41-s − 1.96·43-s − 3.06·45-s + 1.43·47-s + 0.142·49-s + 0.616·51-s + 0.657·53-s + 0.198·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - 2.40e3T \) |
| good | 3 | \( 1 - 230.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 2.53e3T + 1.95e6T^{2} \) |
| 11 | \( 1 + 5.30e3T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.38e5T + 1.06e10T^{2} \) |
| 17 | \( 1 - 1.29e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 8.86e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.67e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 4.30e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 2.81e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.40e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.56e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 4.41e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 4.79e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 3.77e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 2.30e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 7.39e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.59e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 4.39e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.04e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 3.20e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 2.06e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 2.25e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 9.23e8T + 7.60e17T^{2} \) |
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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
−12.74047563024446519190747963765, −11.81145767165269648862618861883, −10.28555940574249049890103859814, −8.783623609013820541425871189909, −7.975730834183554272743420552319, −7.26698523355313289315319394109, −4.48541597234023404536912806159, −3.55816888269928965253486492442, −2.19459673883279359069617379080, 0,
2.19459673883279359069617379080, 3.55816888269928965253486492442, 4.48541597234023404536912806159, 7.26698523355313289315319394109, 7.975730834183554272743420552319, 8.783623609013820541425871189909, 10.28555940574249049890103859814, 11.81145767165269648862618861883, 12.74047563024446519190747963765
