| L(s) = 1 | + (−265. + 459. i)2-s + (9.52e3 + 1.64e4i)3-s + (−7.50e4 − 1.29e5i)4-s + (2.24e5 − 3.88e5i)5-s − 1.00e7·6-s + (−1.05e7 − 1.10e7i)7-s + 1.00e7·8-s + (−1.16e8 + 2.02e8i)9-s + (1.19e8 + 2.06e8i)10-s + (−5.29e8 − 9.16e8i)11-s + (1.42e9 − 2.47e9i)12-s − 1.79e9·13-s + (7.86e9 − 1.89e9i)14-s + 8.55e9·15-s + (7.16e9 − 1.24e10i)16-s + (2.07e9 + 3.60e9i)17-s + ⋯ |
| L(s) = 1 | + (−0.732 + 1.26i)2-s + (0.838 + 1.45i)3-s + (−0.572 − 0.991i)4-s + (0.257 − 0.445i)5-s − 2.45·6-s + (−0.689 − 0.724i)7-s + 0.212·8-s + (−0.904 + 1.56i)9-s + (0.376 + 0.651i)10-s + (−0.744 − 1.28i)11-s + (0.959 − 1.66i)12-s − 0.608·13-s + (1.42 − 0.343i)14-s + 0.861·15-s + (0.417 − 0.722i)16-s + (0.0723 + 0.125i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.295 + 0.955i)\, \overline{\Lambda}(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (0.295 + 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(\approx\) |
\(0.253047 - 0.186578i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.253047 - 0.186578i\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (1.05e7 + 1.10e7i)T \) |
| good | 2 | \( 1 + (265. - 459. i)T + (-6.55e4 - 1.13e5i)T^{2} \) |
| 3 | \( 1 + (-9.52e3 - 1.64e4i)T + (-6.45e7 + 1.11e8i)T^{2} \) |
| 5 | \( 1 + (-2.24e5 + 3.88e5i)T + (-3.81e11 - 6.60e11i)T^{2} \) |
| 11 | \( 1 + (5.29e8 + 9.16e8i)T + (-2.52e17 + 4.37e17i)T^{2} \) |
| 13 | \( 1 + 1.79e9T + 8.65e18T^{2} \) |
| 17 | \( 1 + (-2.07e9 - 3.60e9i)T + (-4.13e20 + 7.16e20i)T^{2} \) |
| 19 | \( 1 + (4.53e10 - 7.85e10i)T + (-2.74e21 - 4.74e21i)T^{2} \) |
| 23 | \( 1 + (3.17e11 - 5.50e11i)T + (-7.05e22 - 1.22e23i)T^{2} \) |
| 29 | \( 1 + 1.62e12T + 7.25e24T^{2} \) |
| 31 | \( 1 + (1.43e12 + 2.48e12i)T + (-1.12e25 + 1.95e25i)T^{2} \) |
| 37 | \( 1 + (-3.61e11 + 6.25e11i)T + (-2.28e26 - 3.95e26i)T^{2} \) |
| 41 | \( 1 + 8.70e13T + 2.61e27T^{2} \) |
| 43 | \( 1 - 3.31e13T + 5.87e27T^{2} \) |
| 47 | \( 1 + (-3.10e13 + 5.38e13i)T + (-1.33e28 - 2.30e28i)T^{2} \) |
| 53 | \( 1 + (-3.11e14 - 5.40e14i)T + (-1.02e29 + 1.77e29i)T^{2} \) |
| 59 | \( 1 + (-2.18e14 - 3.78e14i)T + (-6.35e29 + 1.10e30i)T^{2} \) |
| 61 | \( 1 + (-9.27e14 + 1.60e15i)T + (-1.12e30 - 1.94e30i)T^{2} \) |
| 67 | \( 1 + (3.75e14 + 6.50e14i)T + (-5.52e30 + 9.56e30i)T^{2} \) |
| 71 | \( 1 + 3.64e15T + 2.96e31T^{2} \) |
| 73 | \( 1 + (2.44e15 + 4.23e15i)T + (-2.37e31 + 4.11e31i)T^{2} \) |
| 79 | \( 1 + (-8.44e15 + 1.46e16i)T + (-9.09e31 - 1.57e32i)T^{2} \) |
| 83 | \( 1 - 1.74e16T + 4.21e32T^{2} \) |
| 89 | \( 1 + (2.22e16 - 3.85e16i)T + (-6.89e32 - 1.19e33i)T^{2} \) |
| 97 | \( 1 + 9.07e16T + 5.95e33T^{2} \) |
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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
−19.08503140077857481900535864263, −16.89419256078521954468161293606, −16.27320983494450635034779418026, −15.13323921637153715322826627864, −13.71512157945217018321106236760, −10.26564311958015051621747107795, −9.190061564311257205543207034703, −7.901400229413651515581082638751, −5.57498643543891402187466752276, −3.50368927561926455500465437518,
0.14156735927688682853869849342, 2.13597976932978289928035699400, 2.64469175065879286483477094566, 6.82771410675958726772746968907, 8.597247392378902131473543902606, 10.05942152115049796872732759982, 12.20256239325383340090996062528, 12.95711488182070595676832162224, 14.86612860266634006387233006499, 17.88389174799590502504244396266
