| L(s) = 1 | − 168.·2-s + 6.56e3·3-s − 1.02e5·4-s − 1.10e6·6-s + 2.46e6·7-s + 3.94e7·8-s + 4.30e7·9-s − 9.18e8·11-s − 6.73e8·12-s − 5.59e9·13-s − 4.16e8·14-s + 6.81e9·16-s + 1.89e10·17-s − 7.25e9·18-s − 1.07e11·19-s + 1.61e10·21-s + 1.54e11·22-s + 6.17e11·23-s + 2.58e11·24-s + 9.42e11·26-s + 2.82e11·27-s − 2.53e11·28-s − 3.24e11·29-s − 7.66e12·31-s − 6.31e12·32-s − 6.02e12·33-s − 3.19e12·34-s + ⋯ |
| L(s) = 1 | − 0.465·2-s + 0.577·3-s − 0.783·4-s − 0.268·6-s + 0.161·7-s + 0.830·8-s + 0.333·9-s − 1.29·11-s − 0.452·12-s − 1.90·13-s − 0.0753·14-s + 0.396·16-s + 0.659·17-s − 0.155·18-s − 1.45·19-s + 0.0934·21-s + 0.601·22-s + 1.64·23-s + 0.479·24-s + 0.885·26-s + 0.192·27-s − 0.126·28-s − 0.120·29-s − 1.61·31-s − 1.01·32-s − 0.746·33-s − 0.307·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(\approx\) |
\(0.8313435616\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8313435616\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 6.56e3T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 168.T + 1.31e5T^{2} \) |
| 7 | \( 1 - 2.46e6T + 2.32e14T^{2} \) |
| 11 | \( 1 + 9.18e8T + 5.05e17T^{2} \) |
| 13 | \( 1 + 5.59e9T + 8.65e18T^{2} \) |
| 17 | \( 1 - 1.89e10T + 8.27e20T^{2} \) |
| 19 | \( 1 + 1.07e11T + 5.48e21T^{2} \) |
| 23 | \( 1 - 6.17e11T + 1.41e23T^{2} \) |
| 29 | \( 1 + 3.24e11T + 7.25e24T^{2} \) |
| 31 | \( 1 + 7.66e12T + 2.25e25T^{2} \) |
| 37 | \( 1 - 1.16e13T + 4.56e26T^{2} \) |
| 41 | \( 1 - 1.26e13T + 2.61e27T^{2} \) |
| 43 | \( 1 - 6.34e13T + 5.87e27T^{2} \) |
| 47 | \( 1 + 2.95e14T + 2.66e28T^{2} \) |
| 53 | \( 1 - 5.71e14T + 2.05e29T^{2} \) |
| 59 | \( 1 + 1.31e15T + 1.27e30T^{2} \) |
| 61 | \( 1 - 1.00e14T + 2.24e30T^{2} \) |
| 67 | \( 1 + 1.83e15T + 1.10e31T^{2} \) |
| 71 | \( 1 - 5.00e15T + 2.96e31T^{2} \) |
| 73 | \( 1 + 2.35e15T + 4.74e31T^{2} \) |
| 79 | \( 1 + 2.06e16T + 1.81e32T^{2} \) |
| 83 | \( 1 - 1.09e16T + 4.21e32T^{2} \) |
| 89 | \( 1 - 4.23e16T + 1.37e33T^{2} \) |
| 97 | \( 1 + 9.31e16T + 5.95e33T^{2} \) |
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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
−10.83663230365405249449131460680, −9.933886204624995573444281164963, −9.057156250029137351971917974802, −7.970324846076253566652721646647, −7.26641756490864347580743997042, −5.28331567994007245661552597844, −4.52046155791379987427356913783, −3.01789388283301511958756546690, −1.92332600035384662740606084077, −0.41859752817226026926272815716,
0.41859752817226026926272815716, 1.92332600035384662740606084077, 3.01789388283301511958756546690, 4.52046155791379987427356913783, 5.28331567994007245661552597844, 7.26641756490864347580743997042, 7.970324846076253566652721646647, 9.057156250029137351971917974802, 9.933886204624995573444281164963, 10.83663230365405249449131460680
