| L(s) = 1 | − 12.2·2-s − 362.·4-s + 4.34e3·7-s + 1.06e4·8-s − 3.67e4·11-s + 1.87e5·13-s − 5.30e4·14-s + 5.50e4·16-s + 3.94e3·17-s − 2.37e5·19-s + 4.49e5·22-s + 2.19e6·23-s − 2.29e6·26-s − 1.57e6·28-s + 6.47e6·29-s − 4.92e6·31-s − 6.14e6·32-s − 4.81e4·34-s + 6.61e6·37-s + 2.89e6·38-s − 2.22e7·41-s + 1.25e7·43-s + 1.33e7·44-s − 2.67e7·46-s − 3.42e7·47-s − 2.14e7·49-s − 6.80e7·52-s + ⋯ |
| L(s) = 1 | − 0.540·2-s − 0.708·4-s + 0.683·7-s + 0.922·8-s − 0.756·11-s + 1.82·13-s − 0.369·14-s + 0.210·16-s + 0.0114·17-s − 0.417·19-s + 0.408·22-s + 1.63·23-s − 0.983·26-s − 0.484·28-s + 1.70·29-s − 0.957·31-s − 1.03·32-s − 0.00618·34-s + 0.580·37-s + 0.225·38-s − 1.22·41-s + 0.557·43-s + 0.536·44-s − 0.881·46-s − 1.02·47-s − 0.532·49-s − 1.28·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.574037800\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.574037800\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 12.2T + 512T^{2} \) |
| 7 | \( 1 - 4.34e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 3.67e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.87e5T + 1.06e10T^{2} \) |
| 17 | \( 1 - 3.94e3T + 1.18e11T^{2} \) |
| 19 | \( 1 + 2.37e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 2.19e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 6.47e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 4.92e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 6.61e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 2.22e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.25e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 3.42e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 3.09e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 7.50e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.30e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.37e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.12e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 2.65e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 2.69e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 5.22e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 3.29e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.12e9T + 7.60e17T^{2} \) |
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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
−10.72801199555240978410343419263, −9.525622457838628768623687199829, −8.487386242841374210375010902961, −8.114368576185101031539979492644, −6.72637016074206998332222346499, −5.37168466998929960798680358683, −4.49216914509691840709037334163, −3.24770887955911828910721223498, −1.58758374434184062309557544418, −0.69627322067405062860873231768,
0.69627322067405062860873231768, 1.58758374434184062309557544418, 3.24770887955911828910721223498, 4.49216914509691840709037334163, 5.37168466998929960798680358683, 6.72637016074206998332222346499, 8.114368576185101031539979492644, 8.487386242841374210375010902961, 9.525622457838628768623687199829, 10.72801199555240978410343419263
