| L(s) = 1 | + (2.85 + 4.94i)2-s + (−366. + 634. i)3-s + (1.00e3 − 1.74e3i)4-s + (−4.49e3 − 7.77e3i)5-s − 4.18e3·6-s + (−4.30e4 + 1.12e4i)7-s + 2.31e4·8-s + (−1.79e5 − 3.11e5i)9-s + (2.56e4 − 4.44e4i)10-s + (−4.56e4 + 7.91e4i)11-s + (7.37e5 + 1.27e6i)12-s − 7.75e5·13-s + (−1.78e5 − 1.80e5i)14-s + 6.57e6·15-s + (−1.99e6 − 3.45e6i)16-s + (−4.63e6 + 8.03e6i)17-s + ⋯ |
| L(s) = 1 | + (0.0630 + 0.109i)2-s + (−0.869 + 1.50i)3-s + (0.492 − 0.852i)4-s + (−0.642 − 1.11i)5-s − 0.219·6-s + (−0.967 + 0.253i)7-s + 0.250·8-s + (−1.01 − 1.75i)9-s + (0.0810 − 0.140i)10-s + (−0.0855 + 0.148i)11-s + (0.856 + 1.48i)12-s − 0.579·13-s + (−0.0887 − 0.0896i)14-s + 2.23·15-s + (−0.476 − 0.824i)16-s + (−0.792 + 1.37i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.754 + 0.656i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.754 + 0.656i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.0487260 - 0.130130i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0487260 - 0.130130i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (4.30e4 - 1.12e4i)T \) |
| good | 2 | \( 1 + (-2.85 - 4.94i)T + (-1.02e3 + 1.77e3i)T^{2} \) |
| 3 | \( 1 + (366. - 634. i)T + (-8.85e4 - 1.53e5i)T^{2} \) |
| 5 | \( 1 + (4.49e3 + 7.77e3i)T + (-2.44e7 + 4.22e7i)T^{2} \) |
| 11 | \( 1 + (4.56e4 - 7.91e4i)T + (-1.42e11 - 2.47e11i)T^{2} \) |
| 13 | \( 1 + 7.75e5T + 1.79e12T^{2} \) |
| 17 | \( 1 + (4.63e6 - 8.03e6i)T + (-1.71e13 - 2.96e13i)T^{2} \) |
| 19 | \( 1 + (-1.43e6 - 2.48e6i)T + (-5.82e13 + 1.00e14i)T^{2} \) |
| 23 | \( 1 + (5.26e6 + 9.12e6i)T + (-4.76e14 + 8.25e14i)T^{2} \) |
| 29 | \( 1 + 4.24e7T + 1.22e16T^{2} \) |
| 31 | \( 1 + (-5.29e7 + 9.16e7i)T + (-1.27e16 - 2.20e16i)T^{2} \) |
| 37 | \( 1 + (1.54e8 + 2.68e8i)T + (-8.89e16 + 1.54e17i)T^{2} \) |
| 41 | \( 1 + 1.19e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 1.16e9T + 9.29e17T^{2} \) |
| 47 | \( 1 + (-1.25e8 - 2.16e8i)T + (-1.23e18 + 2.14e18i)T^{2} \) |
| 53 | \( 1 + (1.73e9 - 3.00e9i)T + (-4.63e18 - 8.02e18i)T^{2} \) |
| 59 | \( 1 + (3.01e9 - 5.22e9i)T + (-1.50e19 - 2.61e19i)T^{2} \) |
| 61 | \( 1 + (5.31e9 + 9.21e9i)T + (-2.17e19 + 3.76e19i)T^{2} \) |
| 67 | \( 1 + (1.10e9 - 1.91e9i)T + (-6.10e19 - 1.05e20i)T^{2} \) |
| 71 | \( 1 + 1.34e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + (-8.02e9 + 1.38e10i)T + (-1.56e20 - 2.71e20i)T^{2} \) |
| 79 | \( 1 + (8.41e9 + 1.45e10i)T + (-3.73e20 + 6.47e20i)T^{2} \) |
| 83 | \( 1 + 1.25e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + (-3.02e10 - 5.23e10i)T + (-1.38e21 + 2.40e21i)T^{2} \) |
| 97 | \( 1 + 1.39e11T + 7.15e21T^{2} \) |
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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
−19.51764309041193800488070345804, −16.93149809068399174266704570756, −15.96054883049609375098211601348, −15.18335947428569046092628743032, −12.28667337512325470899101928523, −10.67440489914489058892614703211, −9.341153434044794574582844018643, −5.90300302620653240857006996460, −4.39223036092373458183256995366, −0.085473558361817323349221169540,
2.78676592299824409132422704096, 6.71542105758421729121066238703, 7.41231965281054532513636615392, 11.12595471639540128244153248417, 12.15706636633424500420268475843, 13.49291136794178079755422149239, 15.99201363410612209447787117383, 17.46383531083045124380705695107, 18.72199073788921829830687765013, 19.81870455446407048321117954170
