L(s) = 1 | − 2.82·2-s − 5.19·3-s + 8.00·4-s + 30.1i·5-s + 14.6·6-s + 56.6i·7-s − 22.6·8-s + 27·9-s − 85.3i·10-s − 36.1i·11-s − 41.5·12-s + 133.·13-s − 160. i·14-s − 156. i·15-s + 64.0·16-s + 66.1i·17-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.500·4-s + 1.20i·5-s + 0.408·6-s + 1.15i·7-s − 0.353·8-s + 0.333·9-s − 0.853i·10-s − 0.298i·11-s − 0.288·12-s + 0.789·13-s − 0.818i·14-s − 0.697i·15-s + 0.250·16-s + 0.228i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0541i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0541i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.0148856 + 0.549841i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0148856 + 0.549841i\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 2.82T \) |
| 3 | \( 1 + 5.19T \) |
| 23 | \( 1 + (528. + 28.6i)T \) |
good | 5 | \( 1 - 30.1iT - 625T^{2} \) |
| 7 | \( 1 - 56.6iT - 2.40e3T^{2} \) |
| 11 | \( 1 + 36.1iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 133.T + 2.85e4T^{2} \) |
| 17 | \( 1 - 66.1iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 298. iT - 1.30e5T^{2} \) |
| 29 | \( 1 + 913.T + 7.07e5T^{2} \) |
| 31 | \( 1 + 571.T + 9.23e5T^{2} \) |
| 37 | \( 1 + 821. iT - 1.87e6T^{2} \) |
| 41 | \( 1 + 1.67e3T + 2.82e6T^{2} \) |
| 43 | \( 1 - 395. iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 1.09e3T + 4.87e6T^{2} \) |
| 53 | \( 1 - 2.75e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 1.07e3T + 1.21e7T^{2} \) |
| 61 | \( 1 + 5.88e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 - 2.37e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 3.27e3T + 2.54e7T^{2} \) |
| 73 | \( 1 - 6.20e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 426. iT - 3.89e7T^{2} \) |
| 83 | \( 1 + 1.60e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 1.36e4iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 9.81e3iT - 8.85e7T^{2} \) |
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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
−12.68489748737781823047674574446, −11.65370533296786269371082579957, −10.94138587051392167151043067625, −10.03906381387373653184412610446, −8.846792553138662028062826692075, −7.68826531278210074173112118946, −6.39466842132799805988811621270, −5.71278281129170377118162152967, −3.47436357965158825487236540536, −1.95386461946138755303533298285,
0.31199494381736314259877208393, 1.46116192743396777635584728906, 3.98018475129137012250476605443, 5.23687965304948264595993511514, 6.66691944750064032763242344852, 7.76894080629097216583939197677, 8.883011590885236682566861928283, 9.904300399246238280689016985444, 10.89544138240148267987278182609, 11.82833869071428839388071383616