L(s) = 1 | − 9.69·2-s + 4.57·3-s + 61.9·4-s − 25·5-s − 44.3·6-s + 180.·7-s − 290.·8-s − 222.·9-s + 242.·10-s + 121·11-s + 283.·12-s + 102.·13-s − 1.74e3·14-s − 114.·15-s + 830.·16-s + 638.·17-s + 2.15e3·18-s + 361·19-s − 1.54e3·20-s + 826.·21-s − 1.17e3·22-s − 2.49e3·23-s − 1.32e3·24-s + 625·25-s − 996.·26-s − 2.12e3·27-s + 1.11e4·28-s + ⋯ |
L(s) = 1 | − 1.71·2-s + 0.293·3-s + 1.93·4-s − 0.447·5-s − 0.502·6-s + 1.39·7-s − 1.60·8-s − 0.913·9-s + 0.766·10-s + 0.301·11-s + 0.568·12-s + 0.168·13-s − 2.38·14-s − 0.131·15-s + 0.811·16-s + 0.536·17-s + 1.56·18-s + 0.229·19-s − 0.865·20-s + 0.408·21-s − 0.516·22-s − 0.984·23-s − 0.470·24-s + 0.200·25-s − 0.289·26-s − 0.561·27-s + 2.69·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + 25T \) |
| 11 | \( 1 - 121T \) |
| 19 | \( 1 - 361T \) |
good | 2 | \( 1 + 9.69T + 32T^{2} \) |
| 3 | \( 1 - 4.57T + 243T^{2} \) |
| 7 | \( 1 - 180.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 102.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 638.T + 1.41e6T^{2} \) |
| 23 | \( 1 + 2.49e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.36e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 7.22e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 5.41e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.47e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 128.T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.30e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 713.T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.03e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.06e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.94e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 7.32e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.98e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.23e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.60e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 4.72e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 6.74e3T + 8.58e9T^{2} \) |
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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
−8.663276561342522464876341025025, −7.966978116452586486539513214748, −7.76962013975640789182807543994, −6.57564998830941347122451420309, −5.53489992073446369406321407606, −4.31913109456753267659295729606, −2.99487897938518148603172458824, −1.93334019843296089041711280285, −1.10061647148334606747424286207, 0,
1.10061647148334606747424286207, 1.93334019843296089041711280285, 2.99487897938518148603172458824, 4.31913109456753267659295729606, 5.53489992073446369406321407606, 6.57564998830941347122451420309, 7.76962013975640789182807543994, 7.966978116452586486539513214748, 8.663276561342522464876341025025