| L(s) = 1 | − 3-s − 0.305·5-s − 0.140·7-s + 9-s + 1.85·11-s − 1.94·13-s + 0.305·15-s + 5.66·17-s − 1.04·19-s + 0.140·21-s + 23-s − 4.90·25-s − 27-s − 29-s − 6.72·31-s − 1.85·33-s + 0.0428·35-s − 1.42·37-s + 1.94·39-s + 0.522·41-s + 4.27·43-s − 0.305·45-s + 7.07·47-s − 6.98·49-s − 5.66·51-s − 11.0·53-s − 0.568·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.136·5-s − 0.0529·7-s + 0.333·9-s + 0.560·11-s − 0.538·13-s + 0.0789·15-s + 1.37·17-s − 0.240·19-s + 0.0305·21-s + 0.208·23-s − 0.981·25-s − 0.192·27-s − 0.185·29-s − 1.20·31-s − 0.323·33-s + 0.00723·35-s − 0.234·37-s + 0.311·39-s + 0.0816·41-s + 0.652·43-s − 0.0455·45-s + 1.03·47-s − 0.997·49-s − 0.793·51-s − 1.51·53-s − 0.0766·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 5 | \( 1 + 0.305T + 5T^{2} \) |
| 7 | \( 1 + 0.140T + 7T^{2} \) |
| 11 | \( 1 - 1.85T + 11T^{2} \) |
| 13 | \( 1 + 1.94T + 13T^{2} \) |
| 17 | \( 1 - 5.66T + 17T^{2} \) |
| 19 | \( 1 + 1.04T + 19T^{2} \) |
| 31 | \( 1 + 6.72T + 31T^{2} \) |
| 37 | \( 1 + 1.42T + 37T^{2} \) |
| 41 | \( 1 - 0.522T + 41T^{2} \) |
| 43 | \( 1 - 4.27T + 43T^{2} \) |
| 47 | \( 1 - 7.07T + 47T^{2} \) |
| 53 | \( 1 + 11.0T + 53T^{2} \) |
| 59 | \( 1 + 10.0T + 59T^{2} \) |
| 61 | \( 1 - 7.67T + 61T^{2} \) |
| 67 | \( 1 + 12.0T + 67T^{2} \) |
| 71 | \( 1 - 8.29T + 71T^{2} \) |
| 73 | \( 1 + 0.0686T + 73T^{2} \) |
| 79 | \( 1 - 8.58T + 79T^{2} \) |
| 83 | \( 1 - 7.65T + 83T^{2} \) |
| 89 | \( 1 - 8.93T + 89T^{2} \) |
| 97 | \( 1 - 4.43T + 97T^{2} \) |
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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
−7.63832302604901244375852517683, −6.74438561877811091472797996440, −6.06390539679480133066993507395, −5.44862837420902125625548307256, −4.75132834337730233409644747293, −3.88755068121190086244519408659, −3.26630463919963835991186350133, −2.10309108356817580903630521657, −1.19367625317729728681209256683, 0,
1.19367625317729728681209256683, 2.10309108356817580903630521657, 3.26630463919963835991186350133, 3.88755068121190086244519408659, 4.75132834337730233409644747293, 5.44862837420902125625548307256, 6.06390539679480133066993507395, 6.74438561877811091472797996440, 7.63832302604901244375852517683
