| L(s) = 1 | + 13.8·2-s − 68.3·3-s − 319.·4-s − 768.·5-s − 948.·6-s + 1.12e4·7-s − 1.15e4·8-s − 1.50e4·9-s − 1.06e4·10-s + 2.18e4·12-s − 1.91e5·13-s + 1.56e5·14-s + 5.25e4·15-s + 3.44e3·16-s + 2.63e5·17-s − 2.08e5·18-s − 2.56e5·19-s + 2.45e5·20-s − 7.70e5·21-s − 6.12e4·23-s + 7.88e5·24-s − 1.36e6·25-s − 2.65e6·26-s + 2.37e6·27-s − 3.60e6·28-s − 6.10e6·29-s + 7.28e5·30-s + ⋯ |
| L(s) = 1 | + 0.613·2-s − 0.487·3-s − 0.623·4-s − 0.550·5-s − 0.298·6-s + 1.77·7-s − 0.995·8-s − 0.762·9-s − 0.337·10-s + 0.303·12-s − 1.85·13-s + 1.08·14-s + 0.267·15-s + 0.0131·16-s + 0.764·17-s − 0.467·18-s − 0.451·19-s + 0.343·20-s − 0.864·21-s − 0.0456·23-s + 0.485·24-s − 0.697·25-s − 1.13·26-s + 0.858·27-s − 1.10·28-s − 1.60·29-s + 0.164·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.310006671\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.310006671\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| good | 2 | \( 1 - 13.8T + 512T^{2} \) |
| 3 | \( 1 + 68.3T + 1.96e4T^{2} \) |
| 5 | \( 1 + 768.T + 1.95e6T^{2} \) |
| 7 | \( 1 - 1.12e4T + 4.03e7T^{2} \) |
| 13 | \( 1 + 1.91e5T + 1.06e10T^{2} \) |
| 17 | \( 1 - 2.63e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 2.56e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 6.12e4T + 1.80e12T^{2} \) |
| 29 | \( 1 + 6.10e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 1.84e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 8.27e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.15e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 7.83e6T + 5.02e14T^{2} \) |
| 47 | \( 1 - 1.66e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 5.98e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 5.86e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.08e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.16e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.52e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 2.46e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 5.52e7T + 1.19e17T^{2} \) |
| 83 | \( 1 - 3.69e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 4.85e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 2.85e8T + 7.60e17T^{2} \) |
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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
−11.83765214586485915691445597424, −11.03676346053371754744871217850, −9.589924927087905754027010236424, −8.307142978481847618262388036167, −7.53434168378799965750030064473, −5.63709470795670040960448715079, −4.99129945260115658208907155761, −4.03864982976315207688670708248, −2.36491020697347494387896446696, −0.56084166033006726531484871599,
0.56084166033006726531484871599, 2.36491020697347494387896446696, 4.03864982976315207688670708248, 4.99129945260115658208907155761, 5.63709470795670040960448715079, 7.53434168378799965750030064473, 8.307142978481847618262388036167, 9.589924927087905754027010236424, 11.03676346053371754744871217850, 11.83765214586485915691445597424
