| L(s) = 1 | + (32 − 55.4i)2-s + (746. + 1.29e3i)3-s + (−2.04e3 − 3.54e3i)4-s + (7.17e3 − 1.24e4i)5-s + 9.55e4·6-s − 2.62e5·8-s + (−3.18e5 + 5.51e5i)9-s + (−4.59e5 − 7.95e5i)10-s + (2.14e6 + 3.71e6i)11-s + (3.05e6 − 5.29e6i)12-s − 5.50e6·13-s + 2.14e7·15-s + (−8.38e6 + 1.45e7i)16-s + (−1.98e7 − 3.43e7i)17-s + (2.03e7 + 3.52e7i)18-s + (2.25e7 − 3.89e7i)19-s + ⋯ |
| L(s) = 1 | + (0.353 − 0.612i)2-s + (0.591 + 1.02i)3-s + (−0.249 − 0.433i)4-s + (0.205 − 0.355i)5-s + 0.836·6-s − 0.353·8-s + (−0.199 + 0.345i)9-s + (−0.145 − 0.251i)10-s + (0.364 + 0.631i)11-s + (0.295 − 0.512i)12-s − 0.316·13-s + 0.486·15-s + (−0.125 + 0.216i)16-s + (−0.199 − 0.345i)17-s + (0.141 + 0.244i)18-s + (0.109 − 0.190i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(3.051829898\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.051829898\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-32 + 55.4i)T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + (-746. - 1.29e3i)T + (-7.97e5 + 1.38e6i)T^{2} \) |
| 5 | \( 1 + (-7.17e3 + 1.24e4i)T + (-6.10e8 - 1.05e9i)T^{2} \) |
| 11 | \( 1 + (-2.14e6 - 3.71e6i)T + (-1.72e13 + 2.98e13i)T^{2} \) |
| 13 | \( 1 + 5.50e6T + 3.02e14T^{2} \) |
| 17 | \( 1 + (1.98e7 + 3.43e7i)T + (-4.95e15 + 8.57e15i)T^{2} \) |
| 19 | \( 1 + (-2.25e7 + 3.89e7i)T + (-2.10e16 - 3.64e16i)T^{2} \) |
| 23 | \( 1 + (3.77e8 - 6.53e8i)T + (-2.52e17 - 4.36e17i)T^{2} \) |
| 29 | \( 1 + 2.46e9T + 1.02e19T^{2} \) |
| 31 | \( 1 + (-3.06e9 - 5.31e9i)T + (-1.22e19 + 2.11e19i)T^{2} \) |
| 37 | \( 1 + (-9.52e9 + 1.64e10i)T + (-1.21e20 - 2.10e20i)T^{2} \) |
| 41 | \( 1 - 4.23e10T + 9.25e20T^{2} \) |
| 43 | \( 1 + 7.42e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + (-4.03e10 + 6.98e10i)T + (-2.73e21 - 4.72e21i)T^{2} \) |
| 53 | \( 1 + (-1.10e11 - 1.91e11i)T + (-1.30e22 + 2.25e22i)T^{2} \) |
| 59 | \( 1 + (-2.29e11 - 3.98e11i)T + (-5.24e22 + 9.09e22i)T^{2} \) |
| 61 | \( 1 + (-3.23e10 + 5.61e10i)T + (-8.09e22 - 1.40e23i)T^{2} \) |
| 67 | \( 1 + (-2.07e11 - 3.58e11i)T + (-2.74e23 + 4.74e23i)T^{2} \) |
| 71 | \( 1 + 6.38e11T + 1.16e24T^{2} \) |
| 73 | \( 1 + (-2.92e11 - 5.07e11i)T + (-8.35e23 + 1.44e24i)T^{2} \) |
| 79 | \( 1 + (1.22e12 - 2.12e12i)T + (-2.33e24 - 4.04e24i)T^{2} \) |
| 83 | \( 1 + 3.32e12T + 8.87e24T^{2} \) |
| 89 | \( 1 + (1.39e12 - 2.41e12i)T + (-1.09e25 - 1.90e25i)T^{2} \) |
| 97 | \( 1 - 7.80e12T + 6.73e25T^{2} \) |
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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
−11.48574941763667199480156965585, −10.26931407825814426065716609805, −9.506088672173752113380989823672, −8.804075723402023093474522730489, −7.16794159662495576203544598171, −5.49409822869236530713944618546, −4.48611299771363515255803423198, −3.61314817436273774442732312341, −2.43393140588448153105992816960, −1.13544695586186724191787291553,
0.55663501721014662955280169075, 1.99011090017779131204682904065, 3.01322482592077238090545226607, 4.42619916207515627914241735745, 6.01603626808333862762685146919, 6.76532445988506957900910234679, 7.86770202117416278879338222662, 8.590033814364191975338266495807, 10.00438924162258260786753944418, 11.45728342694214391883104691513
