| L(s) = 1 | − 4.09e3·2-s + 1.22e6·3-s + 1.67e7·4-s − 5.00e9·6-s − 6.80e10·7-s − 6.87e10·8-s + 6.46e11·9-s − 1.63e13·11-s + 2.05e13·12-s − 1.23e14·13-s + 2.78e14·14-s + 2.81e14·16-s − 1.24e15·17-s − 2.65e15·18-s + 1.08e16·19-s − 8.32e16·21-s + 6.69e16·22-s − 1.26e17·23-s − 8.40e16·24-s + 5.04e17·26-s − 2.44e17·27-s − 1.14e18·28-s − 1.19e18·29-s + 3.33e18·31-s − 1.15e18·32-s − 1.99e19·33-s + 5.09e18·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.32·3-s + 0.5·4-s − 0.939·6-s − 1.85·7-s − 0.353·8-s + 0.763·9-s − 1.57·11-s + 0.664·12-s − 1.46·13-s + 1.31·14-s + 0.250·16-s − 0.517·17-s − 0.539·18-s + 1.12·19-s − 2.46·21-s + 1.11·22-s − 1.20·23-s − 0.469·24-s + 1.03·26-s − 0.313·27-s − 0.929·28-s − 0.627·29-s + 0.761·31-s − 0.176·32-s − 2.08·33-s + 0.365·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+25/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(13)\) |
\(\approx\) |
\(0.4103607786\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4103607786\) |
| \(L(\frac{27}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4.09e3T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 1.22e6T + 8.47e11T^{2} \) |
| 7 | \( 1 + 6.80e10T + 1.34e21T^{2} \) |
| 11 | \( 1 + 1.63e13T + 1.08e26T^{2} \) |
| 13 | \( 1 + 1.23e14T + 7.05e27T^{2} \) |
| 17 | \( 1 + 1.24e15T + 5.77e30T^{2} \) |
| 19 | \( 1 - 1.08e16T + 9.30e31T^{2} \) |
| 23 | \( 1 + 1.26e17T + 1.10e34T^{2} \) |
| 29 | \( 1 + 1.19e18T + 3.63e36T^{2} \) |
| 31 | \( 1 - 3.33e18T + 1.92e37T^{2} \) |
| 37 | \( 1 + 5.02e19T + 1.60e39T^{2} \) |
| 41 | \( 1 + 2.63e20T + 2.08e40T^{2} \) |
| 43 | \( 1 + 1.36e20T + 6.86e40T^{2} \) |
| 47 | \( 1 - 7.03e20T + 6.34e41T^{2} \) |
| 53 | \( 1 - 3.75e21T + 1.27e43T^{2} \) |
| 59 | \( 1 + 6.89e21T + 1.86e44T^{2} \) |
| 61 | \( 1 - 1.54e22T + 4.29e44T^{2} \) |
| 67 | \( 1 - 8.66e22T + 4.48e45T^{2} \) |
| 71 | \( 1 - 4.41e22T + 1.91e46T^{2} \) |
| 73 | \( 1 - 8.03e22T + 3.82e46T^{2} \) |
| 79 | \( 1 + 5.07e23T + 2.75e47T^{2} \) |
| 83 | \( 1 + 5.22e23T + 9.48e47T^{2} \) |
| 89 | \( 1 + 1.15e24T + 5.42e48T^{2} \) |
| 97 | \( 1 - 4.89e24T + 4.66e49T^{2} \) |
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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
−10.07779148808716821083721977867, −9.875365583390313374556927493767, −8.746577517471506137065185360740, −7.69357849309276165084779013019, −6.89952407811820208669351021222, −5.40719265683779097234132018946, −3.57244058315239107294767432002, −2.76704271504475840933055190327, −2.17119819878770355508022833182, −0.25252090467895608243274312224,
0.25252090467895608243274312224, 2.17119819878770355508022833182, 2.76704271504475840933055190327, 3.57244058315239107294767432002, 5.40719265683779097234132018946, 6.89952407811820208669351021222, 7.69357849309276165084779013019, 8.746577517471506137065185360740, 9.875365583390313374556927493767, 10.07779148808716821083721977867
