| L(s) = 1 | − 0.915·2-s − 8.77·3-s − 7.16·4-s + 8.03·6-s + 13.8·8-s + 49.9·9-s − 45.6·11-s + 62.8·12-s − 16.0·13-s + 44.5·16-s + 7.28·17-s − 45.7·18-s + 75.6·19-s + 41.8·22-s − 15.8·23-s − 121.·24-s + 14.7·26-s − 201.·27-s + 119.·29-s + 59.5·31-s − 151.·32-s + 400.·33-s − 6.67·34-s − 357.·36-s − 304.·37-s − 69.2·38-s + 141.·39-s + ⋯ |
| L(s) = 1 | − 0.323·2-s − 1.68·3-s − 0.895·4-s + 0.546·6-s + 0.613·8-s + 1.85·9-s − 1.25·11-s + 1.51·12-s − 0.343·13-s + 0.696·16-s + 0.103·17-s − 0.599·18-s + 0.913·19-s + 0.405·22-s − 0.143·23-s − 1.03·24-s + 0.111·26-s − 1.43·27-s + 0.764·29-s + 0.345·31-s − 0.839·32-s + 2.11·33-s − 0.0336·34-s − 1.65·36-s − 1.35·37-s − 0.295·38-s + 0.579·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.2553708112\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2553708112\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 0.915T + 8T^{2} \) |
| 3 | \( 1 + 8.77T + 27T^{2} \) |
| 11 | \( 1 + 45.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 16.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 7.28T + 4.91e3T^{2} \) |
| 19 | \( 1 - 75.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + 15.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 119.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 59.5T + 2.97e4T^{2} \) |
| 37 | \( 1 + 304.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 238.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 365.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 100.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 740.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 285.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 400.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 128.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 39.1T + 3.57e5T^{2} \) |
| 73 | \( 1 + 961.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 552.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 33.6T + 5.71e5T^{2} \) |
| 89 | \( 1 - 856.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 771.T + 9.12e5T^{2} \) |
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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
−9.670596033934037869558627683029, −8.488537605778158335548270284079, −7.68262039875955515929687413573, −6.82797063036917393026725702417, −5.79068778318718189288028155772, −5.04598011662329551835821072085, −4.69060612358520639276087991973, −3.26906649466948532804918315785, −1.47812967960477643445960167529, −0.31531534801067252471518172766,
0.31531534801067252471518172766, 1.47812967960477643445960167529, 3.26906649466948532804918315785, 4.69060612358520639276087991973, 5.04598011662329551835821072085, 5.79068778318718189288028155772, 6.82797063036917393026725702417, 7.68262039875955515929687413573, 8.488537605778158335548270284079, 9.670596033934037869558627683029
