| L(s) = 1 | − 5.36·2-s + 20.7·4-s + 1.66·5-s − 7·7-s − 68.4·8-s − 8.94·10-s − 11·11-s + 0.996·13-s + 37.5·14-s + 200.·16-s + 99.5·17-s + 32.8·19-s + 34.6·20-s + 58.9·22-s − 72.6·23-s − 122.·25-s − 5.34·26-s − 145.·28-s − 45.0·29-s − 62.8·31-s − 529.·32-s − 533.·34-s − 11.6·35-s − 301.·37-s − 176.·38-s − 114.·40-s + 307.·41-s + ⋯ |
| L(s) = 1 | − 1.89·2-s + 2.59·4-s + 0.149·5-s − 0.377·7-s − 3.02·8-s − 0.282·10-s − 0.301·11-s + 0.0212·13-s + 0.716·14-s + 3.13·16-s + 1.41·17-s + 0.396·19-s + 0.387·20-s + 0.571·22-s − 0.658·23-s − 0.977·25-s − 0.0403·26-s − 0.980·28-s − 0.288·29-s − 0.363·31-s − 2.92·32-s − 2.69·34-s − 0.0564·35-s − 1.33·37-s − 0.752·38-s − 0.451·40-s + 1.17·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 + 5.36T + 8T^{2} \) |
| 5 | \( 1 - 1.66T + 125T^{2} \) |
| 13 | \( 1 - 0.996T + 2.19e3T^{2} \) |
| 17 | \( 1 - 99.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 32.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 72.6T + 1.21e4T^{2} \) |
| 29 | \( 1 + 45.0T + 2.43e4T^{2} \) |
| 31 | \( 1 + 62.8T + 2.97e4T^{2} \) |
| 37 | \( 1 + 301.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 307.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 214.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 602.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 592.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 695.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 442.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 555.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 153.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 147.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 676.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 222.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.13e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.01e3T + 9.12e5T^{2} \) |
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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
−9.548924076862037214516285576207, −8.983872051563959548037896704647, −7.77133634996702908922959108597, −7.55907751947258218193185389336, −6.31350680118817512899238979074, −5.56380282597534844236109311538, −3.55644139608032578899851950203, −2.40070490076710911561856235022, −1.24082907130985927531488781561, 0,
1.24082907130985927531488781561, 2.40070490076710911561856235022, 3.55644139608032578899851950203, 5.56380282597534844236109311538, 6.31350680118817512899238979074, 7.55907751947258218193185389336, 7.77133634996702908922959108597, 8.983872051563959548037896704647, 9.548924076862037214516285576207
