| L(s) = 1 | + (−58.4 − 101. i)2-s + (−1.00e4 + 1.73e4i)3-s + (5.87e4 − 1.01e5i)4-s + (5.26e5 + 9.11e5i)5-s + 2.34e6·6-s + (1.15e7 + 9.94e6i)7-s − 2.90e7·8-s + (−1.37e8 − 2.37e8i)9-s + (6.14e7 − 1.06e8i)10-s + (−3.23e8 + 5.60e8i)11-s + (1.17e9 + 2.04e9i)12-s − 2.11e9·13-s + (3.31e8 − 1.75e9i)14-s − 2.11e10·15-s + (−6.00e9 − 1.03e10i)16-s + (−1.27e10 + 2.21e10i)17-s + ⋯ |
| L(s) = 1 | + (−0.161 − 0.279i)2-s + (−0.883 + 1.53i)3-s + (0.447 − 0.775i)4-s + (0.602 + 1.04i)5-s + 0.570·6-s + (0.757 + 0.652i)7-s − 0.611·8-s + (−1.06 − 1.83i)9-s + (0.194 − 0.336i)10-s + (−0.454 + 0.788i)11-s + (0.791 + 1.37i)12-s − 0.717·13-s + (0.0599 − 0.316i)14-s − 2.13·15-s + (−0.349 − 0.604i)16-s + (−0.444 + 0.769i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 - 0.247i)\, \overline{\Lambda}(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (-0.968 - 0.247i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(\approx\) |
\(0.112839 + 0.896457i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.112839 + 0.896457i\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-1.15e7 - 9.94e6i)T \) |
| good | 2 | \( 1 + (58.4 + 101. i)T + (-6.55e4 + 1.13e5i)T^{2} \) |
| 3 | \( 1 + (1.00e4 - 1.73e4i)T + (-6.45e7 - 1.11e8i)T^{2} \) |
| 5 | \( 1 + (-5.26e5 - 9.11e5i)T + (-3.81e11 + 6.60e11i)T^{2} \) |
| 11 | \( 1 + (3.23e8 - 5.60e8i)T + (-2.52e17 - 4.37e17i)T^{2} \) |
| 13 | \( 1 + 2.11e9T + 8.65e18T^{2} \) |
| 17 | \( 1 + (1.27e10 - 2.21e10i)T + (-4.13e20 - 7.16e20i)T^{2} \) |
| 19 | \( 1 + (2.09e10 + 3.63e10i)T + (-2.74e21 + 4.74e21i)T^{2} \) |
| 23 | \( 1 + (3.20e10 + 5.55e10i)T + (-7.05e22 + 1.22e23i)T^{2} \) |
| 29 | \( 1 + 3.14e12T + 7.25e24T^{2} \) |
| 31 | \( 1 + (-2.92e12 + 5.06e12i)T + (-1.12e25 - 1.95e25i)T^{2} \) |
| 37 | \( 1 + (-1.83e13 - 3.17e13i)T + (-2.28e26 + 3.95e26i)T^{2} \) |
| 41 | \( 1 + 3.62e13T + 2.61e27T^{2} \) |
| 43 | \( 1 + 1.06e14T + 5.87e27T^{2} \) |
| 47 | \( 1 + (-1.23e14 - 2.13e14i)T + (-1.33e28 + 2.30e28i)T^{2} \) |
| 53 | \( 1 + (-7.99e13 + 1.38e14i)T + (-1.02e29 - 1.77e29i)T^{2} \) |
| 59 | \( 1 + (2.66e14 - 4.62e14i)T + (-6.35e29 - 1.10e30i)T^{2} \) |
| 61 | \( 1 + (-6.97e14 - 1.20e15i)T + (-1.12e30 + 1.94e30i)T^{2} \) |
| 67 | \( 1 + (-1.11e15 + 1.93e15i)T + (-5.52e30 - 9.56e30i)T^{2} \) |
| 71 | \( 1 - 7.46e15T + 2.96e31T^{2} \) |
| 73 | \( 1 + (-9.28e14 + 1.60e15i)T + (-2.37e31 - 4.11e31i)T^{2} \) |
| 79 | \( 1 + (4.02e15 + 6.96e15i)T + (-9.09e31 + 1.57e32i)T^{2} \) |
| 83 | \( 1 - 1.23e15T + 4.21e32T^{2} \) |
| 89 | \( 1 + (-7.01e15 - 1.21e16i)T + (-6.89e32 + 1.19e33i)T^{2} \) |
| 97 | \( 1 - 1.70e16T + 5.95e33T^{2} \) |
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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
−18.43906847625975258971340658443, −17.24155503562357953627224303342, −15.20827553889335337151738442498, −14.91811489156774431383123776946, −11.56779255842339170169735654980, −10.55803532392187115743607679414, −9.658967572609868586197907150347, −6.21987375358359015523511417763, −4.89803256995899444862118891517, −2.36649127538812413929653697093,
0.42430753547731199209974587052, 1.92682081580235896219923778087, 5.38836922815733887920168896871, 7.07067103823664330989999564255, 8.285403816472003129855707019567, 11.32316058814269928125833291331, 12.55646055011136679634712470324, 13.60487668390485033644610255720, 16.57451864948172370570570745542, 17.21646054210461044540982757139
