| L(s) = 1 | − 1.24·3-s + 5·5-s + 19.3·7-s − 25.4·9-s + 2.63·11-s + 36.8·13-s − 6.22·15-s + 6.64·17-s − 151.·19-s − 24.1·21-s − 23·23-s + 25·25-s + 65.3·27-s + 132.·29-s + 120.·31-s − 3.27·33-s + 96.9·35-s − 356.·37-s − 45.9·39-s − 341.·41-s + 233.·43-s − 127.·45-s − 345.·47-s + 33.2·49-s − 8.28·51-s − 663.·53-s + 13.1·55-s + ⋯ |
| L(s) = 1 | − 0.239·3-s + 0.447·5-s + 1.04·7-s − 0.942·9-s + 0.0721·11-s + 0.787·13-s − 0.107·15-s + 0.0948·17-s − 1.82·19-s − 0.250·21-s − 0.208·23-s + 0.200·25-s + 0.465·27-s + 0.847·29-s + 0.696·31-s − 0.0172·33-s + 0.468·35-s − 1.58·37-s − 0.188·39-s − 1.30·41-s + 0.827·43-s − 0.421·45-s − 1.07·47-s + 0.0968·49-s − 0.0227·51-s − 1.71·53-s + 0.0322·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 23 | \( 1 + 23T \) |
| good | 3 | \( 1 + 1.24T + 27T^{2} \) |
| 7 | \( 1 - 19.3T + 343T^{2} \) |
| 11 | \( 1 - 2.63T + 1.33e3T^{2} \) |
| 13 | \( 1 - 36.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 6.64T + 4.91e3T^{2} \) |
| 19 | \( 1 + 151.T + 6.85e3T^{2} \) |
| 29 | \( 1 - 132.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 120.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 356.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 341.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 233.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 345.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 663.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 855.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 449.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 225.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 70.3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 832.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 771.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 304.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 774.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.07e3T + 9.12e5T^{2} \) |
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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
−8.467399470415606871709599554695, −8.000435485325107116111814201239, −6.63626290514227654727682498830, −6.18208321779086527025189648416, −5.20957513228909161339715249344, −4.55258971013084237755821616964, −3.40711454686268640115499631417, −2.26227960690176485913989862338, −1.38126207585986401204065343885, 0,
1.38126207585986401204065343885, 2.26227960690176485913989862338, 3.40711454686268640115499631417, 4.55258971013084237755821616964, 5.20957513228909161339715249344, 6.18208321779086527025189648416, 6.63626290514227654727682498830, 8.000435485325107116111814201239, 8.467399470415606871709599554695
