| L(s) = 1 | + 3-s + 0.509·5-s + 1.30·7-s + 9-s + 3.43·11-s − 4.04·13-s + 0.509·15-s − 3.71·17-s − 1.69·19-s + 1.30·21-s − 23-s − 4.74·25-s + 27-s − 29-s − 5.08·31-s + 3.43·33-s + 0.665·35-s + 6.50·37-s − 4.04·39-s − 5.79·41-s − 2.51·43-s + 0.509·45-s − 10.8·47-s − 5.29·49-s − 3.71·51-s − 8.24·53-s + 1.74·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.227·5-s + 0.494·7-s + 0.333·9-s + 1.03·11-s − 1.12·13-s + 0.131·15-s − 0.900·17-s − 0.388·19-s + 0.285·21-s − 0.208·23-s − 0.948·25-s + 0.192·27-s − 0.185·29-s − 0.913·31-s + 0.597·33-s + 0.112·35-s + 1.06·37-s − 0.647·39-s − 0.904·41-s − 0.383·43-s + 0.0759·45-s − 1.58·47-s − 0.755·49-s − 0.520·51-s − 1.13·53-s + 0.235·55-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 | \( 1 - 0.509T + 5T^{2} \) |
| 7 | \( 1 - 1.30T + 7T^{2} \) |
| 11 | \( 1 - 3.43T + 11T^{2} \) |
| 13 | \( 1 + 4.04T + 13T^{2} \) |
| 17 | \( 1 + 3.71T + 17T^{2} \) |
| 19 | \( 1 + 1.69T + 19T^{2} \) |
| 31 | \( 1 + 5.08T + 31T^{2} \) |
| 37 | \( 1 - 6.50T + 37T^{2} \) |
| 41 | \( 1 + 5.79T + 41T^{2} \) |
| 43 | \( 1 + 2.51T + 43T^{2} \) |
| 47 | \( 1 + 10.8T + 47T^{2} \) |
| 53 | \( 1 + 8.24T + 53T^{2} \) |
| 59 | \( 1 + 2.44T + 59T^{2} \) |
| 61 | \( 1 - 4.30T + 61T^{2} \) |
| 67 | \( 1 + 6.06T + 67T^{2} \) |
| 71 | \( 1 - 0.193T + 71T^{2} \) |
| 73 | \( 1 - 10.7T + 73T^{2} \) |
| 79 | \( 1 + 7.99T + 79T^{2} \) |
| 83 | \( 1 + 5.51T + 83T^{2} \) |
| 89 | \( 1 - 3.60T + 89T^{2} \) |
| 97 | \( 1 + 1.88T + 97T^{2} \) |
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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
−7.60973731516624228538200668152, −6.76167898223043717832863386208, −6.30439690174694706982806588525, −5.27498987072501474183989683724, −4.58085946498589692880442504065, −3.96136056041874838570922435396, −3.07248986836135406030404147797, −2.08050498228254451332047179033, −1.59674913674560803299615554436, 0,
1.59674913674560803299615554436, 2.08050498228254451332047179033, 3.07248986836135406030404147797, 3.96136056041874838570922435396, 4.58085946498589692880442504065, 5.27498987072501474183989683724, 6.30439690174694706982806588525, 6.76167898223043717832863386208, 7.60973731516624228538200668152
