L(s) = 1 | + (−1.22 − 0.707i)2-s + (−1.69 + 2.47i)3-s + (0.999 + 1.73i)4-s + (4.69 + 8.13i)5-s + (3.82 − 1.83i)6-s + (5.67 − 9.82i)7-s − 2.82i·8-s + (−3.24 − 8.39i)9-s − 13.2i·10-s + (3.82 − 6.62i)11-s + (−5.98 − 0.462i)12-s + (15.6 − 9.04i)13-s + (−13.8 + 8.02i)14-s + (−28.0 − 2.17i)15-s + (−2.00 + 3.46i)16-s + 21.1·17-s + ⋯ |
L(s) = 1 | + (−0.612 − 0.353i)2-s + (−0.565 + 0.824i)3-s + (0.249 + 0.433i)4-s + (0.938 + 1.62i)5-s + (0.637 − 0.305i)6-s + (0.810 − 1.40i)7-s − 0.353i·8-s + (−0.360 − 0.932i)9-s − 1.32i·10-s + (0.347 − 0.601i)11-s + (−0.498 − 0.0385i)12-s + (1.20 − 0.695i)13-s + (−0.992 + 0.572i)14-s + (−1.87 − 0.144i)15-s + (−0.125 + 0.216i)16-s + 1.24·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 - 0.199i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.979 - 0.199i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.40326 + 0.141533i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.40326 + 0.141533i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.22 + 0.707i)T \) |
| 3 | \( 1 + (1.69 - 2.47i)T \) |
| 19 | \( 1 + (0.118 + 18.9i)T \) |
good | 5 | \( 1 + (-4.69 - 8.13i)T + (-12.5 + 21.6i)T^{2} \) |
| 7 | \( 1 + (-5.67 + 9.82i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (-3.82 + 6.62i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (-15.6 + 9.04i)T + (84.5 - 146. i)T^{2} \) |
| 17 | \( 1 - 21.1T + 289T^{2} \) |
| 23 | \( 1 + (8.37 + 14.5i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (-12.3 - 7.13i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (15.1 - 8.74i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + 10.6iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-1.55 + 0.896i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-16.7 + 28.9i)T + (-924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (10.3 - 17.8i)T + (-1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 - 7.82iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-66.1 + 38.1i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (17.7 - 30.8i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (96.1 - 55.5i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 54.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 117.T + 5.32e3T^{2} \) |
| 79 | \( 1 + (-14.5 - 8.41i)T + (3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (68.7 - 119. i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 - 7.70iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-101. - 58.4i)T + (4.70e3 + 8.14e3i)T^{2} \) |
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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
−10.91573004846553517161297227514, −10.52664652366945795321288984948, −9.978315578968830564233129425004, −8.736946444062340339504642868731, −7.45772136521905234901005563226, −6.54380424323076113892226337921, −5.59166131193578477146899263068, −3.91860227084301688719974138675, −3.03987223598587780194602533205, −1.06761977133131620151907270490,
1.37849916217219311307342706723, 1.83961578162414092088365851375, 4.75471195724986577462855848738, 5.77590989494944997273206581970, 6.02346720935144861319875289453, 7.75113742352444551365540581129, 8.505565373659804373737490112622, 9.140901554729866222037627322498, 10.12611432964969022965464309357, 11.65215242020750507529352380086