| L(s) = 1 | + 0.210·3-s − 5-s − 2.32·7-s − 2.95·9-s + 0.534·13-s − 0.210·15-s − 2.42·17-s + 4.95·19-s − 0.489·21-s + 4.53·23-s + 25-s − 1.25·27-s + 5.48·29-s − 1.04·31-s + 2.32·35-s + 7.48·37-s + 0.112·39-s − 10.6·41-s + 4.32·43-s + 2.95·45-s − 6.76·47-s − 1.60·49-s − 0.510·51-s − 4.53·53-s + 1.04·57-s + 6.44·59-s − 6.79·61-s + ⋯ |
| L(s) = 1 | + 0.121·3-s − 0.447·5-s − 0.878·7-s − 0.985·9-s + 0.148·13-s − 0.0544·15-s − 0.587·17-s + 1.13·19-s − 0.106·21-s + 0.945·23-s + 0.200·25-s − 0.241·27-s + 1.01·29-s − 0.187·31-s + 0.392·35-s + 1.23·37-s + 0.0180·39-s − 1.65·41-s + 0.659·43-s + 0.440·45-s − 0.987·47-s − 0.228·49-s − 0.0714·51-s − 0.622·53-s + 0.138·57-s + 0.839·59-s − 0.870·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 0.210T + 3T^{2} \) |
| 7 | \( 1 + 2.32T + 7T^{2} \) |
| 13 | \( 1 - 0.534T + 13T^{2} \) |
| 17 | \( 1 + 2.42T + 17T^{2} \) |
| 19 | \( 1 - 4.95T + 19T^{2} \) |
| 23 | \( 1 - 4.53T + 23T^{2} \) |
| 29 | \( 1 - 5.48T + 29T^{2} \) |
| 31 | \( 1 + 1.04T + 31T^{2} \) |
| 37 | \( 1 - 7.48T + 37T^{2} \) |
| 41 | \( 1 + 10.6T + 41T^{2} \) |
| 43 | \( 1 - 4.32T + 43T^{2} \) |
| 47 | \( 1 + 6.76T + 47T^{2} \) |
| 53 | \( 1 + 4.53T + 53T^{2} \) |
| 59 | \( 1 - 6.44T + 59T^{2} \) |
| 61 | \( 1 + 6.79T + 61T^{2} \) |
| 67 | \( 1 + 0.721T + 67T^{2} \) |
| 71 | \( 1 + 4.53T + 71T^{2} \) |
| 73 | \( 1 + 1.06T + 73T^{2} \) |
| 79 | \( 1 - 4.64T + 79T^{2} \) |
| 83 | \( 1 - 13.6T + 83T^{2} \) |
| 89 | \( 1 - 12.7T + 89T^{2} \) |
| 97 | \( 1 - 4.75T + 97T^{2} \) |
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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
−7.37352250221532850222630487488, −6.55144593964437420343801274241, −6.16686349375918283803193296662, −5.19931695280512682329121898204, −4.66459711896080910277114399454, −3.51977384182201143068259465136, −3.17021522492174310863378054237, −2.39423408564807677525988068937, −1.05333166926606518011081583007, 0,
1.05333166926606518011081583007, 2.39423408564807677525988068937, 3.17021522492174310863378054237, 3.51977384182201143068259465136, 4.66459711896080910277114399454, 5.19931695280512682329121898204, 6.16686349375918283803193296662, 6.55144593964437420343801274241, 7.37352250221532850222630487488
