| L(s) = 1 | + (−3.81 + 6.60i)2-s + (−187. + 324. i)3-s + (4.06e3 + 7.04e3i)4-s + 4.24e4·5-s + (−1.42e3 − 2.47e3i)6-s + (−1.78e5 − 3.09e5i)7-s − 1.24e5·8-s + (7.27e5 + 1.25e6i)9-s + (−1.61e5 + 2.80e5i)10-s + (−4.82e5 + 8.35e5i)11-s − 3.04e6·12-s + (1.54e7 + 7.95e6i)13-s + 2.72e6·14-s + (−7.94e6 + 1.37e7i)15-s + (−3.28e7 + 5.68e7i)16-s + (3.98e7 + 6.89e7i)17-s + ⋯ |
| L(s) = 1 | + (−0.0421 + 0.0729i)2-s + (−0.148 + 0.256i)3-s + (0.496 + 0.859i)4-s + 1.21·5-s + (−0.0124 − 0.0216i)6-s + (−0.573 − 0.993i)7-s − 0.167·8-s + (0.456 + 0.789i)9-s + (−0.0511 + 0.0885i)10-s + (−0.0821 + 0.142i)11-s − 0.294·12-s + (0.889 + 0.456i)13-s + 0.0967·14-s + (−0.180 + 0.311i)15-s + (−0.489 + 0.847i)16-s + (0.399 + 0.692i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.217 - 0.976i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (0.217 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(1.71376 + 1.37402i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.71376 + 1.37402i\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (-1.54e7 - 7.95e6i)T \) |
| good | 2 | \( 1 + (3.81 - 6.60i)T + (-4.09e3 - 7.09e3i)T^{2} \) |
| 3 | \( 1 + (187. - 324. i)T + (-7.97e5 - 1.38e6i)T^{2} \) |
| 5 | \( 1 - 4.24e4T + 1.22e9T^{2} \) |
| 7 | \( 1 + (1.78e5 + 3.09e5i)T + (-4.84e10 + 8.39e10i)T^{2} \) |
| 11 | \( 1 + (4.82e5 - 8.35e5i)T + (-1.72e13 - 2.98e13i)T^{2} \) |
| 17 | \( 1 + (-3.98e7 - 6.89e7i)T + (-4.95e15 + 8.57e15i)T^{2} \) |
| 19 | \( 1 + (-3.00e7 - 5.20e7i)T + (-2.10e16 + 3.64e16i)T^{2} \) |
| 23 | \( 1 + (5.04e8 - 8.73e8i)T + (-2.52e17 - 4.36e17i)T^{2} \) |
| 29 | \( 1 + (-5.23e8 + 9.06e8i)T + (-5.13e18 - 8.88e18i)T^{2} \) |
| 31 | \( 1 + 9.83e8T + 2.44e19T^{2} \) |
| 37 | \( 1 + (-1.30e10 + 2.25e10i)T + (-1.21e20 - 2.10e20i)T^{2} \) |
| 41 | \( 1 + (-9.88e9 + 1.71e10i)T + (-4.62e20 - 8.01e20i)T^{2} \) |
| 43 | \( 1 + (-3.79e10 - 6.57e10i)T + (-8.59e20 + 1.48e21i)T^{2} \) |
| 47 | \( 1 + 8.31e10T + 5.46e21T^{2} \) |
| 53 | \( 1 - 2.84e11T + 2.60e22T^{2} \) |
| 59 | \( 1 + (2.39e11 + 4.14e11i)T + (-5.24e22 + 9.09e22i)T^{2} \) |
| 61 | \( 1 + (2.06e11 + 3.58e11i)T + (-8.09e22 + 1.40e23i)T^{2} \) |
| 67 | \( 1 + (-1.99e11 + 3.44e11i)T + (-2.74e23 - 4.74e23i)T^{2} \) |
| 71 | \( 1 + (1.33e11 + 2.31e11i)T + (-5.82e23 + 1.00e24i)T^{2} \) |
| 73 | \( 1 - 6.91e11T + 1.67e24T^{2} \) |
| 79 | \( 1 + 3.42e11T + 4.66e24T^{2} \) |
| 83 | \( 1 - 3.40e12T + 8.87e24T^{2} \) |
| 89 | \( 1 + (-1.97e12 + 3.41e12i)T + (-1.09e25 - 1.90e25i)T^{2} \) |
| 97 | \( 1 + (3.37e12 + 5.85e12i)T + (-3.36e25 + 5.82e25i)T^{2} \) |
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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
−16.79343705811317848468391046936, −16.00699161144403306127112072985, −13.81362883784561523707403227917, −12.92110957464714499651775889905, −10.94694193344034583010873524554, −9.705795689805909295999196529993, −7.65338286944614474551075546382, −6.10559753090519641586329553443, −3.83557990967433266646825643242, −1.82259372597462229094334654318,
1.00126672654311416580776032030, 2.55893269258495312975605880778, 5.67590168044102189907883892123, 6.47029964867966133590076308943, 9.194135771128290605702236694176, 10.26589024822361227788032758724, 12.04500575715217317715406630544, 13.49854975503033157227469994753, 14.98948065751672920837317252915, 16.17956149468208460492048590733
