| L(s) = 1 | + 2.17·2-s + 2.71·4-s − 2.18·5-s + 3.77·7-s + 1.54·8-s − 4.75·10-s − 11-s − 2.80·13-s + 8.20·14-s − 2.06·16-s − 0.740·17-s − 0.533·19-s − 5.93·20-s − 2.17·22-s − 2.10·23-s − 0.208·25-s − 6.09·26-s + 10.2·28-s − 8.30·29-s − 3.15·31-s − 7.58·32-s − 1.60·34-s − 8.26·35-s − 2.81·37-s − 1.15·38-s − 3.38·40-s − 8.19·41-s + ⋯ |
| L(s) = 1 | + 1.53·2-s + 1.35·4-s − 0.978·5-s + 1.42·7-s + 0.546·8-s − 1.50·10-s − 0.301·11-s − 0.778·13-s + 2.19·14-s − 0.517·16-s − 0.179·17-s − 0.122·19-s − 1.32·20-s − 0.462·22-s − 0.438·23-s − 0.0417·25-s − 1.19·26-s + 1.93·28-s − 1.54·29-s − 0.567·31-s − 1.34·32-s − 0.275·34-s − 1.39·35-s − 0.462·37-s − 0.187·38-s − 0.535·40-s − 1.27·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 - 2.17T + 2T^{2} \) |
| 5 | \( 1 + 2.18T + 5T^{2} \) |
| 7 | \( 1 - 3.77T + 7T^{2} \) |
| 13 | \( 1 + 2.80T + 13T^{2} \) |
| 17 | \( 1 + 0.740T + 17T^{2} \) |
| 19 | \( 1 + 0.533T + 19T^{2} \) |
| 23 | \( 1 + 2.10T + 23T^{2} \) |
| 29 | \( 1 + 8.30T + 29T^{2} \) |
| 31 | \( 1 + 3.15T + 31T^{2} \) |
| 37 | \( 1 + 2.81T + 37T^{2} \) |
| 41 | \( 1 + 8.19T + 41T^{2} \) |
| 43 | \( 1 - 9.36T + 43T^{2} \) |
| 47 | \( 1 - 1.30T + 47T^{2} \) |
| 53 | \( 1 - 4.44T + 53T^{2} \) |
| 59 | \( 1 - 0.127T + 59T^{2} \) |
| 67 | \( 1 + 0.175T + 67T^{2} \) |
| 71 | \( 1 - 0.0782T + 71T^{2} \) |
| 73 | \( 1 + 0.0568T + 73T^{2} \) |
| 79 | \( 1 + 10.5T + 79T^{2} \) |
| 83 | \( 1 + 4.37T + 83T^{2} \) |
| 89 | \( 1 + 10.9T + 89T^{2} \) |
| 97 | \( 1 - 4.47T + 97T^{2} \) |
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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
−7.38393128264417620257784688450, −7.21083071184774367895995445097, −5.95924760180383865079116474501, −5.35377028400868551550168299883, −4.75679166577398843841313961276, −4.13106406397086357285703941952, −3.58212330312792411668382378812, −2.50387630881054782008760339597, −1.75749280291071720437399199608, 0,
1.75749280291071720437399199608, 2.50387630881054782008760339597, 3.58212330312792411668382378812, 4.13106406397086357285703941952, 4.75679166577398843841313961276, 5.35377028400868551550168299883, 5.95924760180383865079116474501, 7.21083071184774367895995445097, 7.38393128264417620257784688450
