| L(s) = 1 | + 3-s + 3.79·5-s + 2.15·7-s + 9-s + 2.54·11-s − 1.95·13-s + 3.79·15-s + 0.224·17-s − 0.224·19-s + 2.15·21-s − 2.82·23-s + 9.42·25-s + 27-s − 2.62·29-s + 1.84·31-s + 2.54·33-s + 8.19·35-s + 5.18·37-s − 1.95·39-s + 5.88·41-s − 10.9·43-s + 3.79·45-s + 2.82·47-s − 2.33·49-s + 0.224·51-s + 10.6·53-s + 9.65·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.69·5-s + 0.816·7-s + 0.333·9-s + 0.766·11-s − 0.542·13-s + 0.980·15-s + 0.0545·17-s − 0.0515·19-s + 0.471·21-s − 0.589·23-s + 1.88·25-s + 0.192·27-s − 0.487·29-s + 0.330·31-s + 0.442·33-s + 1.38·35-s + 0.853·37-s − 0.313·39-s + 0.918·41-s − 1.67·43-s + 0.566·45-s + 0.412·47-s − 0.334·49-s + 0.0314·51-s + 1.45·53-s + 1.30·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.701041845\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.701041845\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| good | 5 | \( 1 - 3.79T + 5T^{2} \) |
| 7 | \( 1 - 2.15T + 7T^{2} \) |
| 11 | \( 1 - 2.54T + 11T^{2} \) |
| 13 | \( 1 + 1.95T + 13T^{2} \) |
| 17 | \( 1 - 0.224T + 17T^{2} \) |
| 19 | \( 1 + 0.224T + 19T^{2} \) |
| 23 | \( 1 + 2.82T + 23T^{2} \) |
| 29 | \( 1 + 2.62T + 29T^{2} \) |
| 31 | \( 1 - 1.84T + 31T^{2} \) |
| 37 | \( 1 - 5.18T + 37T^{2} \) |
| 41 | \( 1 - 5.88T + 41T^{2} \) |
| 43 | \( 1 + 10.9T + 43T^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 - 10.6T + 53T^{2} \) |
| 59 | \( 1 + 5.65T + 59T^{2} \) |
| 61 | \( 1 - 8.46T + 61T^{2} \) |
| 67 | \( 1 + 14.7T + 67T^{2} \) |
| 71 | \( 1 + 4.31T + 71T^{2} \) |
| 73 | \( 1 - 5.97T + 73T^{2} \) |
| 79 | \( 1 + 15.0T + 79T^{2} \) |
| 83 | \( 1 - 14.3T + 83T^{2} \) |
| 89 | \( 1 + 1.42T + 89T^{2} \) |
| 97 | \( 1 + 16.3T + 97T^{2} \) |
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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
−8.833156092238623792103733196584, −8.063171648893510354436198396491, −7.19805871758379719764315350351, −6.38695699945454559848990832288, −5.68954389710944627044267913818, −4.89871330476596583375859166434, −4.04937066593116514048240905307, −2.80134687702366256094082924920, −2.02424011702061067180305134123, −1.31055387821963451636965326312,
1.31055387821963451636965326312, 2.02424011702061067180305134123, 2.80134687702366256094082924920, 4.04937066593116514048240905307, 4.89871330476596583375859166434, 5.68954389710944627044267913818, 6.38695699945454559848990832288, 7.19805871758379719764315350351, 8.063171648893510354436198396491, 8.833156092238623792103733196584
