L(s) = 1 | − 12.1i·2-s + (−18.4 − 19.7i)3-s − 84.1·4-s − 4.50i·5-s + (−239. + 224. i)6-s + 129.·7-s + 245. i·8-s + (−48.3 + 727. i)9-s − 54.8·10-s − 332. i·11-s + (1.55e3 + 1.65e3i)12-s − 3.03e3·13-s − 1.57e3i·14-s + (−88.8 + 83.1i)15-s − 2.39e3·16-s − 8.74e3i·17-s + ⋯ |
L(s) = 1 | − 1.52i·2-s + (−0.683 − 0.730i)3-s − 1.31·4-s − 0.0360i·5-s + (−1.11 + 1.03i)6-s + 0.377·7-s + 0.479i·8-s + (−0.0662 + 0.997i)9-s − 0.0548·10-s − 0.249i·11-s + (0.898 + 0.960i)12-s − 1.38·13-s − 0.575i·14-s + (−0.0263 + 0.0246i)15-s − 0.585·16-s − 1.78i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.683 - 0.730i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.683 - 0.730i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(0.345540 + 0.796579i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.345540 + 0.796579i\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (18.4 + 19.7i)T \) |
| 7 | \( 1 - 129.T \) |
good | 2 | \( 1 + 12.1iT - 64T^{2} \) |
| 5 | \( 1 + 4.50iT - 1.56e4T^{2} \) |
| 11 | \( 1 + 332. iT - 1.77e6T^{2} \) |
| 13 | \( 1 + 3.03e3T + 4.82e6T^{2} \) |
| 17 | \( 1 + 8.74e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 - 2.52e3T + 4.70e7T^{2} \) |
| 23 | \( 1 - 305. iT - 1.48e8T^{2} \) |
| 29 | \( 1 + 1.59e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 + 4.55e4T + 8.87e8T^{2} \) |
| 37 | \( 1 - 6.19e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + 2.75e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 1.24e5T + 6.32e9T^{2} \) |
| 47 | \( 1 - 2.45e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 + 2.57e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 + 1.58e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 1.15e5T + 5.15e10T^{2} \) |
| 67 | \( 1 + 3.28e5T + 9.04e10T^{2} \) |
| 71 | \( 1 - 3.84e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 2.40e5T + 1.51e11T^{2} \) |
| 79 | \( 1 - 4.40e5T + 2.43e11T^{2} \) |
| 83 | \( 1 - 1.00e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 - 9.61e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 9.73e5T + 8.32e11T^{2} \) |
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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
−16.38972393409448959305300798895, −14.16996208224788468337306882437, −12.87120142845811711456894338796, −11.87529495814664166295363058872, −11.01027650271654721574494093637, −9.511664244507354622218254438873, −7.32023069555729014993812100560, −4.96771715157316624203870195215, −2.44922258492772732693019279014, −0.59708101761326252455495282298,
4.60098212351708496930235408665, 5.90122800610158876267459602792, 7.43379741822243395588459017405, 9.121156056603143120967478279267, 10.74559116156133546126421396122, 12.49699252781275290308790938416, 14.59762349024471527576682638337, 15.07576510392426160328613424905, 16.48635379530151892435478930595, 17.14868752418632570438686352745