| L(s) = 1 | + 0.381·3-s + 3.85·7-s − 2.85·9-s − 11-s − 1.76·13-s − 1.61·17-s − 6.70·19-s + 1.47·21-s + 7.09·23-s − 2.23·27-s − 3.61·29-s + 3·31-s − 0.381·33-s − 5.76·37-s − 0.673·39-s − 3·41-s − 6·43-s − 5.94·47-s + 7.85·49-s − 0.618·51-s + 6.32·53-s − 2.56·57-s − 9.47·59-s − 11.0·61-s − 11.0·63-s + 8·67-s + 2.70·69-s + ⋯ |
| L(s) = 1 | + 0.220·3-s + 1.45·7-s − 0.951·9-s − 0.301·11-s − 0.489·13-s − 0.392·17-s − 1.53·19-s + 0.321·21-s + 1.47·23-s − 0.430·27-s − 0.671·29-s + 0.538·31-s − 0.0664·33-s − 0.947·37-s − 0.107·39-s − 0.468·41-s − 0.914·43-s − 0.867·47-s + 1.12·49-s − 0.0865·51-s + 0.868·53-s − 0.339·57-s − 1.23·59-s − 1.41·61-s − 1.38·63-s + 0.977·67-s + 0.326·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 - 0.381T + 3T^{2} \) |
| 7 | \( 1 - 3.85T + 7T^{2} \) |
| 13 | \( 1 + 1.76T + 13T^{2} \) |
| 17 | \( 1 + 1.61T + 17T^{2} \) |
| 19 | \( 1 + 6.70T + 19T^{2} \) |
| 23 | \( 1 - 7.09T + 23T^{2} \) |
| 29 | \( 1 + 3.61T + 29T^{2} \) |
| 31 | \( 1 - 3T + 31T^{2} \) |
| 37 | \( 1 + 5.76T + 37T^{2} \) |
| 41 | \( 1 + 3T + 41T^{2} \) |
| 43 | \( 1 + 6T + 43T^{2} \) |
| 47 | \( 1 + 5.94T + 47T^{2} \) |
| 53 | \( 1 - 6.32T + 53T^{2} \) |
| 59 | \( 1 + 9.47T + 59T^{2} \) |
| 61 | \( 1 + 11.0T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 - 14.1T + 71T^{2} \) |
| 73 | \( 1 + 12.6T + 73T^{2} \) |
| 79 | \( 1 - 0.854T + 79T^{2} \) |
| 83 | \( 1 + 16.8T + 83T^{2} \) |
| 89 | \( 1 + 18.0T + 89T^{2} \) |
| 97 | \( 1 - 0.618T + 97T^{2} \) |
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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
−8.259741115214459591493335730899, −7.34332128446837249426664486604, −6.62635956897337541844045736723, −5.63953997479103078442159367514, −4.96333895446597968853112499311, −4.41632042810119722171234999991, −3.25357155132373839925404481607, −2.37413644325933034098614993449, −1.58162896782745415033246615935, 0,
1.58162896782745415033246615935, 2.37413644325933034098614993449, 3.25357155132373839925404481607, 4.41632042810119722171234999991, 4.96333895446597968853112499311, 5.63953997479103078442159367514, 6.62635956897337541844045736723, 7.34332128446837249426664486604, 8.259741115214459591493335730899
