| L(s) = 1 | − 5.29e3·2-s + 2.60e5·3-s + 1.96e7·4-s − 1.96e7·5-s − 1.37e9·6-s − 5.94e10·8-s − 2.62e10·9-s + 1.03e11·10-s − 5.52e11·11-s + 5.11e12·12-s − 2.38e12·13-s − 5.11e12·15-s + 1.50e14·16-s − 1.83e14·17-s + 1.38e14·18-s − 7.40e14·19-s − 3.84e14·20-s + 2.92e15·22-s − 8.19e15·23-s − 1.54e16·24-s − 1.15e16·25-s + 1.26e16·26-s − 3.13e16·27-s − 5.35e16·29-s + 2.70e16·30-s − 1.51e17·31-s − 2.95e17·32-s + ⋯ |
| L(s) = 1 | − 1.82·2-s + 0.849·3-s + 2.33·4-s − 0.179·5-s − 1.55·6-s − 2.44·8-s − 0.278·9-s + 0.328·10-s − 0.584·11-s + 1.98·12-s − 0.368·13-s − 0.152·15-s + 2.13·16-s − 1.29·17-s + 0.508·18-s − 1.45·19-s − 0.420·20-s + 1.06·22-s − 1.79·23-s − 2.07·24-s − 0.967·25-s + 0.673·26-s − 1.08·27-s − 0.814·29-s + 0.278·30-s − 1.06·31-s − 1.45·32-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
| \(L(12)\) |
\(\approx\) |
\(0.1097650019\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1097650019\) |
| \(L(\frac{25}{2})\) |
|
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 | \( 1 + 5.29e3T + 8.38e6T^{2} \) |
| 3 | \( 1 - 2.60e5T + 9.41e10T^{2} \) |
| 5 | \( 1 + 1.96e7T + 1.19e16T^{2} \) |
| 11 | \( 1 + 5.52e11T + 8.95e23T^{2} \) |
| 13 | \( 1 + 2.38e12T + 4.17e25T^{2} \) |
| 17 | \( 1 + 1.83e14T + 1.99e28T^{2} \) |
| 19 | \( 1 + 7.40e14T + 2.57e29T^{2} \) |
| 23 | \( 1 + 8.19e15T + 2.08e31T^{2} \) |
| 29 | \( 1 + 5.35e16T + 4.31e33T^{2} \) |
| 31 | \( 1 + 1.51e17T + 2.00e34T^{2} \) |
| 37 | \( 1 - 1.32e17T + 1.17e36T^{2} \) |
| 41 | \( 1 - 1.89e18T + 1.24e37T^{2} \) |
| 43 | \( 1 + 8.74e18T + 3.71e37T^{2} \) |
| 47 | \( 1 - 1.17e19T + 2.87e38T^{2} \) |
| 53 | \( 1 - 1.10e20T + 4.55e39T^{2} \) |
| 59 | \( 1 - 3.28e20T + 5.36e40T^{2} \) |
| 61 | \( 1 + 2.62e19T + 1.15e41T^{2} \) |
| 67 | \( 1 - 2.71e19T + 9.99e41T^{2} \) |
| 71 | \( 1 - 6.33e20T + 3.79e42T^{2} \) |
| 73 | \( 1 + 2.96e21T + 7.18e42T^{2} \) |
| 79 | \( 1 - 2.00e21T + 4.42e43T^{2} \) |
| 83 | \( 1 - 1.11e22T + 1.37e44T^{2} \) |
| 89 | \( 1 + 1.48e22T + 6.85e44T^{2} \) |
| 97 | \( 1 - 6.30e22T + 4.96e45T^{2} \) |
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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
−10.77440765390309079940256688477, −9.784673159739977580164302402147, −8.774751137458741759510752861538, −8.165493001851716315517676061554, −7.23930468577760775585741186565, −5.96535104239947142121110883754, −3.87845028202318793134816294518, −2.27285702255824060468070014888, −2.06272068116150424843256258323, −0.16652376070361260106841417318,
0.16652376070361260106841417318, 2.06272068116150424843256258323, 2.27285702255824060468070014888, 3.87845028202318793134816294518, 5.96535104239947142121110883754, 7.23930468577760775585741186565, 8.165493001851716315517676061554, 8.774751137458741759510752861538, 9.784673159739977580164302402147, 10.77440765390309079940256688477
