| L(s) = 1 | + 2.93e3·2-s − 1.42e5·3-s + 2.18e5·4-s − 6.28e6·5-s − 4.17e8·6-s − 2.39e10·8-s − 7.38e10·9-s − 1.84e10·10-s − 1.02e12·11-s − 3.10e10·12-s − 7.32e12·13-s + 8.94e11·15-s − 7.21e13·16-s − 9.55e13·17-s − 2.16e14·18-s + 1.44e14·19-s − 1.37e12·20-s − 2.99e15·22-s − 2.76e15·23-s + 3.41e15·24-s − 1.18e16·25-s − 2.15e16·26-s + 2.39e16·27-s + 5.32e16·29-s + 2.62e15·30-s − 1.67e17·31-s − 1.06e16·32-s + ⋯ |
| L(s) = 1 | + 1.01·2-s − 0.463·3-s + 0.0260·4-s − 0.0575·5-s − 0.469·6-s − 0.986·8-s − 0.784·9-s − 0.0583·10-s − 1.07·11-s − 0.0120·12-s − 1.13·13-s + 0.0267·15-s − 1.02·16-s − 0.676·17-s − 0.795·18-s + 0.284·19-s − 0.00149·20-s − 1.09·22-s − 0.606·23-s + 0.457·24-s − 0.996·25-s − 1.14·26-s + 0.827·27-s + 0.810·29-s + 0.0270·30-s − 1.18·31-s − 0.0520·32-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
| \(L(12)\) |
\(\approx\) |
\(0.3449152886\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3449152886\) |
| \(L(\frac{25}{2})\) |
|
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 | \( 1 - 2.93e3T + 8.38e6T^{2} \) |
| 3 | \( 1 + 1.42e5T + 9.41e10T^{2} \) |
| 5 | \( 1 + 6.28e6T + 1.19e16T^{2} \) |
| 11 | \( 1 + 1.02e12T + 8.95e23T^{2} \) |
| 13 | \( 1 + 7.32e12T + 4.17e25T^{2} \) |
| 17 | \( 1 + 9.55e13T + 1.99e28T^{2} \) |
| 19 | \( 1 - 1.44e14T + 2.57e29T^{2} \) |
| 23 | \( 1 + 2.76e15T + 2.08e31T^{2} \) |
| 29 | \( 1 - 5.32e16T + 4.31e33T^{2} \) |
| 31 | \( 1 + 1.67e17T + 2.00e34T^{2} \) |
| 37 | \( 1 + 2.19e14T + 1.17e36T^{2} \) |
| 41 | \( 1 + 1.49e18T + 1.24e37T^{2} \) |
| 43 | \( 1 - 3.07e18T + 3.71e37T^{2} \) |
| 47 | \( 1 + 1.45e19T + 2.87e38T^{2} \) |
| 53 | \( 1 - 1.90e19T + 4.55e39T^{2} \) |
| 59 | \( 1 + 1.03e20T + 5.36e40T^{2} \) |
| 61 | \( 1 + 2.58e20T + 1.15e41T^{2} \) |
| 67 | \( 1 + 3.42e20T + 9.99e41T^{2} \) |
| 71 | \( 1 + 2.75e21T + 3.79e42T^{2} \) |
| 73 | \( 1 + 4.44e21T + 7.18e42T^{2} \) |
| 79 | \( 1 - 5.95e21T + 4.42e43T^{2} \) |
| 83 | \( 1 - 3.35e21T + 1.37e44T^{2} \) |
| 89 | \( 1 - 3.56e22T + 6.85e44T^{2} \) |
| 97 | \( 1 - 5.51e22T + 4.96e45T^{2} \) |
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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
−11.51018803471992995354520453021, −10.22153413539704584474727118529, −8.933021234805027449783847027883, −7.63713587936291018173169835887, −6.17953640226464802871269998759, −5.31508103440150175658384746864, −4.53902022120033761488308817258, −3.18688856994799690570708502094, −2.24677142147702191686478991020, −0.20755426634599584004880731165,
0.20755426634599584004880731165, 2.24677142147702191686478991020, 3.18688856994799690570708502094, 4.53902022120033761488308817258, 5.31508103440150175658384746864, 6.17953640226464802871269998759, 7.63713587936291018173169835887, 8.933021234805027449783847027883, 10.22153413539704584474727118529, 11.51018803471992995354520453021
