| L(s) = 1 | − 2.82·2-s + 5.19·3-s + 8.00·4-s + 7.78i·5-s − 14.6·6-s − 25.8i·7-s − 22.6·8-s + 27·9-s − 22.0i·10-s − 131. i·11-s + 41.5·12-s + 98.7·13-s + 72.9i·14-s + 40.4i·15-s + 64.0·16-s + 271. i·17-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.500·4-s + 0.311i·5-s − 0.408·6-s − 0.526i·7-s − 0.353·8-s + 0.333·9-s − 0.220i·10-s − 1.08i·11-s + 0.288·12-s + 0.584·13-s + 0.372i·14-s + 0.179i·15-s + 0.250·16-s + 0.938i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.680 + 0.733i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.680 + 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.42112 - 0.620005i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.42112 - 0.620005i\) |
| \(L(3)\) |
|
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 | \( 1 + (359. + 387. i)T \) |
| good | 5 | \( 1 - 7.78iT - 625T^{2} \) |
| 7 | \( 1 + 25.8iT - 2.40e3T^{2} \) |
| 11 | \( 1 + 131. iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 98.7T + 2.85e4T^{2} \) |
| 17 | \( 1 - 271. iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 342. iT - 1.30e5T^{2} \) |
| 29 | \( 1 - 560.T + 7.07e5T^{2} \) |
| 31 | \( 1 - 1.08e3T + 9.23e5T^{2} \) |
| 37 | \( 1 + 2.35e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 - 1.28e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + 3.25e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 - 1.16e3T + 4.87e6T^{2} \) |
| 53 | \( 1 - 1.59e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 3.66e3T + 1.21e7T^{2} \) |
| 61 | \( 1 - 1.88e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 - 1.31e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 7.24e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + 753.T + 2.83e7T^{2} \) |
| 79 | \( 1 - 3.74e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 - 1.04e4iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 2.62e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 8.54e3iT - 8.85e7T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.30037561884104266775471638585, −10.89105326367482214650167163432, −10.45055873141273098322807707661, −8.985770535570386881280750332887, −8.332210666377041584539626465273, −7.13299617411764545512245119802, −6.03284952295263196860525104752, −4.00972274949631950113428287329, −2.64439888155844314279465244270, −0.828661423770063129617577076967,
1.41687642221805143420716538564, 2.88214121437045072192469025603, 4.64636170528524862187893539782, 6.25813709380208096952024427322, 7.56067092969401784577067311955, 8.469523995169897281402391415801, 9.462478328912609908999090876359, 10.21757982372688473128084012943, 11.67470863281621877129491443602, 12.45121179476547951229978231113
