| L(s) = 1 | − 4.51e3·2-s − 3.53e5·3-s + 1.19e7·4-s − 1.94e8·5-s + 1.59e9·6-s − 1.62e10·8-s + 3.05e10·9-s + 8.76e11·10-s − 1.22e12·11-s − 4.23e12·12-s − 1.36e11·13-s + 6.86e13·15-s − 2.73e13·16-s + 2.11e14·17-s − 1.38e14·18-s − 4.59e14·19-s − 2.32e15·20-s + 5.51e15·22-s + 2.16e15·23-s + 5.72e15·24-s + 2.58e16·25-s + 6.15e14·26-s + 2.24e16·27-s − 3.79e16·29-s − 3.09e17·30-s − 7.62e16·31-s + 2.59e17·32-s + ⋯ |
| L(s) = 1 | − 1.55·2-s − 1.15·3-s + 1.42·4-s − 1.77·5-s + 1.79·6-s − 0.667·8-s + 0.324·9-s + 2.77·10-s − 1.29·11-s − 1.64·12-s − 0.0211·13-s + 2.04·15-s − 0.388·16-s + 1.49·17-s − 0.506·18-s − 0.905·19-s − 2.54·20-s + 2.01·22-s + 0.474·23-s + 0.767·24-s + 2.16·25-s + 0.0329·26-s + 0.777·27-s − 0.577·29-s − 3.19·30-s − 0.539·31-s + 1.27·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+23/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(12)\) |
\(\approx\) |
\(0.09187191019\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09187191019\) |
| \(L(\frac{25}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 + 4.51e3T + 8.38e6T^{2} \) |
| 3 | \( 1 + 3.53e5T + 9.41e10T^{2} \) |
| 5 | \( 1 + 1.94e8T + 1.19e16T^{2} \) |
| 11 | \( 1 + 1.22e12T + 8.95e23T^{2} \) |
| 13 | \( 1 + 1.36e11T + 4.17e25T^{2} \) |
| 17 | \( 1 - 2.11e14T + 1.99e28T^{2} \) |
| 19 | \( 1 + 4.59e14T + 2.57e29T^{2} \) |
| 23 | \( 1 - 2.16e15T + 2.08e31T^{2} \) |
| 29 | \( 1 + 3.79e16T + 4.31e33T^{2} \) |
| 31 | \( 1 + 7.62e16T + 2.00e34T^{2} \) |
| 37 | \( 1 - 4.27e17T + 1.17e36T^{2} \) |
| 41 | \( 1 - 2.13e18T + 1.24e37T^{2} \) |
| 43 | \( 1 - 5.86e17T + 3.71e37T^{2} \) |
| 47 | \( 1 - 2.96e19T + 2.87e38T^{2} \) |
| 53 | \( 1 + 1.73e18T + 4.55e39T^{2} \) |
| 59 | \( 1 - 3.94e19T + 5.36e40T^{2} \) |
| 61 | \( 1 - 6.22e19T + 1.15e41T^{2} \) |
| 67 | \( 1 + 1.13e21T + 9.99e41T^{2} \) |
| 71 | \( 1 + 1.95e21T + 3.79e42T^{2} \) |
| 73 | \( 1 + 4.53e21T + 7.18e42T^{2} \) |
| 79 | \( 1 + 3.96e21T + 4.42e43T^{2} \) |
| 83 | \( 1 + 9.81e21T + 1.37e44T^{2} \) |
| 89 | \( 1 + 2.11e22T + 6.85e44T^{2} \) |
| 97 | \( 1 - 1.15e22T + 4.96e45T^{2} \) |
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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
−10.94410969077338328266162563410, −10.34475754163302079557638810959, −8.738189983524086677710690222109, −7.78559960746691219109418158981, −7.20935066669917466124962808806, −5.64738965172951725860550147382, −4.36147506006957713806571573401, −2.84537186240430434108589758242, −1.07997927148170173299247262398, −0.22110856803945394481163326548,
0.22110856803945394481163326548, 1.07997927148170173299247262398, 2.84537186240430434108589758242, 4.36147506006957713806571573401, 5.64738965172951725860550147382, 7.20935066669917466124962808806, 7.78559960746691219109418158981, 8.738189983524086677710690222109, 10.34475754163302079557638810959, 10.94410969077338328266162563410
