| L(s) = 1 | + (0.910 + 1.08i)2-s + (−0.0398 − 1.73i)3-s + (−0.340 + 1.97i)4-s + (−0.549 + 1.51i)5-s + (1.83 − 1.62i)6-s + (0.984 + 0.173i)7-s + (−2.44 + 1.42i)8-s + (−2.99 + 0.138i)9-s + (−2.13 + 0.780i)10-s + (−3.39 + 1.23i)11-s + (3.42 + 0.511i)12-s + (−5.16 + 4.33i)13-s + (0.709 + 1.22i)14-s + (2.63 + 0.891i)15-s + (−3.76 − 1.34i)16-s + (1.36 − 0.789i)17-s + ⋯ |
| L(s) = 1 | + (0.644 + 0.764i)2-s + (−0.0230 − 0.999i)3-s + (−0.170 + 0.985i)4-s + (−0.245 + 0.675i)5-s + (0.749 − 0.661i)6-s + (0.372 + 0.0656i)7-s + (−0.863 + 0.504i)8-s + (−0.998 + 0.0460i)9-s + (−0.675 + 0.246i)10-s + (−1.02 + 0.372i)11-s + (0.989 + 0.147i)12-s + (−1.43 + 1.20i)13-s + (0.189 + 0.327i)14-s + (0.680 + 0.230i)15-s + (−0.941 − 0.335i)16-s + (0.331 − 0.191i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.873 - 0.487i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.873 - 0.487i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.293590 + 1.12754i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.293590 + 1.12754i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.910 - 1.08i)T \) |
| 3 | \( 1 + (0.0398 + 1.73i)T \) |
| 7 | \( 1 + (-0.984 - 0.173i)T \) |
| good | 5 | \( 1 + (0.549 - 1.51i)T + (-3.83 - 3.21i)T^{2} \) |
| 11 | \( 1 + (3.39 - 1.23i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (5.16 - 4.33i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-1.36 + 0.789i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.09 - 1.20i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.454 + 2.57i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (4.57 - 5.45i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-6.18 + 1.09i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (3.64 + 6.31i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.190 - 0.226i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-3.57 - 9.81i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (1.29 - 7.36i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + 13.5iT - 53T^{2} \) |
| 59 | \( 1 + (-0.397 - 0.144i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (1.19 - 6.76i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-1.69 - 2.01i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (5.38 + 9.33i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (8.21 - 14.2i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.92 + 3.49i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-9.62 - 8.07i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (-2.89 - 1.66i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.78 + 2.10i)T + (74.3 - 62.3i)T^{2} \) |
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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
−11.03227074254893431888505310032, −9.711698652522369253716457278322, −8.643547941615627487781475182168, −7.56485755589551883062819057880, −7.36219706426725134897237582830, −6.50691980556046950050644594433, −5.40894520883497836744982679211, −4.61789507430676742070219447188, −3.09568206853052266337778072918, −2.21709914233824278912512241793,
0.45069892844401446733486212178, 2.51408556268340931907929951429, 3.40714570335700512619235309569, 4.64248436972353557708297088304, 5.13593111765475174482083426283, 5.83081774360145979753260748541, 7.58841698570874355149670145872, 8.451381648966286155573917766400, 9.436725082201483544817575610954, 10.27514750577672667832560092729
