| 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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 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
−8.211753133798532947614505041855, −7.76806691248245018736349187664, −6.81453195240722876311948874632, −6.42562815211529647545208561870, −5.01646346290773042457835324521, −4.16890181375158337253960920374, −3.48963104647977690276947819401, −3.00304434783108564700097698539, −1.38117913227133862270068312960, 0,
1.38117913227133862270068312960, 3.00304434783108564700097698539, 3.48963104647977690276947819401, 4.16890181375158337253960920374, 5.01646346290773042457835324521, 6.42562815211529647545208561870, 6.81453195240722876311948874632, 7.76806691248245018736349187664, 8.211753133798532947614505041855
