| L(s) = 1 | − 2.77·3-s + 14.3·5-s − 16.7·7-s − 19.3·9-s + 10.9·11-s + 13·13-s − 39.6·15-s + 74.6·17-s + 27.8·19-s + 46.4·21-s − 137.·23-s + 79.9·25-s + 128.·27-s − 166.·29-s − 184.·31-s − 30.2·33-s − 240.·35-s + 188.·37-s − 36.0·39-s + 450.·41-s − 456.·43-s − 276.·45-s − 139.·47-s − 61.6·49-s − 206.·51-s − 290.·53-s + 156.·55-s + ⋯ |
| L(s) = 1 | − 0.533·3-s + 1.28·5-s − 0.905·7-s − 0.715·9-s + 0.299·11-s + 0.277·13-s − 0.683·15-s + 1.06·17-s + 0.335·19-s + 0.483·21-s − 1.24·23-s + 0.639·25-s + 0.915·27-s − 1.06·29-s − 1.06·31-s − 0.159·33-s − 1.15·35-s + 0.838·37-s − 0.147·39-s + 1.71·41-s − 1.61·43-s − 0.916·45-s − 0.434·47-s − 0.179·49-s − 0.568·51-s − 0.752·53-s + 0.382·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 - 13T \) |
| good | 3 | \( 1 + 2.77T + 27T^{2} \) |
| 5 | \( 1 - 14.3T + 125T^{2} \) |
| 7 | \( 1 + 16.7T + 343T^{2} \) |
| 11 | \( 1 - 10.9T + 1.33e3T^{2} \) |
| 17 | \( 1 - 74.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 27.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 137.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 166.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 184.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 188.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 450.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 456.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 139.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 290.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 530.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 30.2T + 2.26e5T^{2} \) |
| 67 | \( 1 - 275.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 191.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 859.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 345.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 925.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.17e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 276.T + 9.12e5T^{2} \) |
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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
−9.699161966533209872431477867055, −8.749121589623530528510096087557, −7.61458172548842676941970751040, −6.44023316269378989692994759614, −5.88332149598400480431352359264, −5.34884814522129355674257243789, −3.79467790586929594959321251044, −2.73563150337890594693401693470, −1.48026392733629552604050118012, 0,
1.48026392733629552604050118012, 2.73563150337890594693401693470, 3.79467790586929594959321251044, 5.34884814522129355674257243789, 5.88332149598400480431352359264, 6.44023316269378989692994759614, 7.61458172548842676941970751040, 8.749121589623530528510096087557, 9.699161966533209872431477867055
