| L(s) = 1 | + 2.71·2-s + 5.37·4-s + 2.37·7-s + 9.15·8-s + 2.20·11-s − 2·13-s + 6.44·14-s + 14.1·16-s − 3.22·17-s + 19-s + 5.99·22-s + 1.01·23-s − 5.43·26-s + 12.7·28-s − 1.01·29-s + 4.74·31-s + 20.0·32-s − 8.74·34-s − 10.7·37-s + 2.71·38-s + 5.43·41-s + 11.1·43-s + 11.8·44-s + 2.74·46-s − 4.23·47-s − 1.37·49-s − 10.7·52-s + ⋯ |
| L(s) = 1 | + 1.91·2-s + 2.68·4-s + 0.896·7-s + 3.23·8-s + 0.666·11-s − 0.554·13-s + 1.72·14-s + 3.52·16-s − 0.781·17-s + 0.229·19-s + 1.27·22-s + 0.210·23-s − 1.06·26-s + 2.40·28-s − 0.187·29-s + 0.852·31-s + 3.53·32-s − 1.49·34-s − 1.76·37-s + 0.440·38-s + 0.848·41-s + 1.69·43-s + 1.78·44-s + 0.404·46-s − 0.617·47-s − 0.196·49-s − 1.49·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.069163653\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.069163653\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 2.71T + 2T^{2} \) |
| 7 | \( 1 - 2.37T + 7T^{2} \) |
| 11 | \( 1 - 2.20T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + 3.22T + 17T^{2} \) |
| 23 | \( 1 - 1.01T + 23T^{2} \) |
| 29 | \( 1 + 1.01T + 29T^{2} \) |
| 31 | \( 1 - 4.74T + 31T^{2} \) |
| 37 | \( 1 + 10.7T + 37T^{2} \) |
| 41 | \( 1 - 5.43T + 41T^{2} \) |
| 43 | \( 1 - 11.1T + 43T^{2} \) |
| 47 | \( 1 + 4.23T + 47T^{2} \) |
| 53 | \( 1 - 9.84T + 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 + 5.11T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 2.02T + 71T^{2} \) |
| 73 | \( 1 - 5.11T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 11.8T + 83T^{2} \) |
| 89 | \( 1 + 9.84T + 89T^{2} \) |
| 97 | \( 1 + 7.48T + 97T^{2} \) |
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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
−8.096478780754545826442068103993, −7.28543296770869624348521719863, −6.79266140651162364451612632082, −5.95246750070678135053446123399, −5.28177288860000187406576089339, −4.54599532496041508820666990428, −4.10954196088245437901309916767, −3.09648093233906247320946157074, −2.27132020531464096992493905499, −1.40655745527787418380906670269,
1.40655745527787418380906670269, 2.27132020531464096992493905499, 3.09648093233906247320946157074, 4.10954196088245437901309916767, 4.54599532496041508820666990428, 5.28177288860000187406576089339, 5.95246750070678135053446123399, 6.79266140651162364451612632082, 7.28543296770869624348521719863, 8.096478780754545826442068103993
