| L(s) = 1 | + (0.776 + 0.0406i)2-s + (−0.706 − 0.872i)3-s + (−1.38 − 0.145i)4-s + (2.15 − 0.589i)5-s + (−0.513 − 0.706i)6-s + (2.33 − 1.24i)7-s + (−2.60 − 0.412i)8-s + (0.361 − 1.70i)9-s + (1.69 − 0.370i)10-s + (1.29 − 0.276i)11-s + (0.853 + 1.31i)12-s + (−1.79 − 0.912i)13-s + (1.86 − 0.870i)14-s + (−2.03 − 1.46i)15-s + (0.724 + 0.153i)16-s + (0.333 + 0.868i)17-s + ⋯ |
| L(s) = 1 | + (0.548 + 0.0287i)2-s + (−0.408 − 0.503i)3-s + (−0.694 − 0.0729i)4-s + (0.964 − 0.263i)5-s + (−0.209 − 0.288i)6-s + (0.882 − 0.470i)7-s + (−0.921 − 0.145i)8-s + (0.120 − 0.566i)9-s + (0.536 − 0.117i)10-s + (0.391 − 0.0832i)11-s + (0.246 + 0.379i)12-s + (−0.496 − 0.253i)13-s + (0.497 − 0.232i)14-s + (−0.526 − 0.378i)15-s + (0.181 + 0.0384i)16-s + (0.0808 + 0.210i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.584 + 0.811i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.584 + 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.19656 - 0.612623i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.19656 - 0.612623i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-2.15 + 0.589i)T \) |
| 7 | \( 1 + (-2.33 + 1.24i)T \) |
| good | 2 | \( 1 + (-0.776 - 0.0406i)T + (1.98 + 0.209i)T^{2} \) |
| 3 | \( 1 + (0.706 + 0.872i)T + (-0.623 + 2.93i)T^{2} \) |
| 11 | \( 1 + (-1.29 + 0.276i)T + (10.0 - 4.47i)T^{2} \) |
| 13 | \( 1 + (1.79 + 0.912i)T + (7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (-0.333 - 0.868i)T + (-12.6 + 11.3i)T^{2} \) |
| 19 | \( 1 + (-0.550 - 5.24i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (0.0599 - 1.14i)T + (-22.8 - 2.40i)T^{2} \) |
| 29 | \( 1 + (4.04 - 5.56i)T + (-8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (3.98 - 8.95i)T + (-20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (-6.42 + 4.17i)T + (15.0 - 33.8i)T^{2} \) |
| 41 | \( 1 + (0.0622 - 0.0202i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (3.34 + 3.34i)T + 43iT^{2} \) |
| 47 | \( 1 + (0.174 + 0.0668i)T + (34.9 + 31.4i)T^{2} \) |
| 53 | \( 1 + (-8.68 + 7.02i)T + (11.0 - 51.8i)T^{2} \) |
| 59 | \( 1 + (3.22 - 3.58i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + (3.39 - 3.06i)T + (6.37 - 60.6i)T^{2} \) |
| 67 | \( 1 + (7.93 - 3.04i)T + (49.7 - 44.8i)T^{2} \) |
| 71 | \( 1 + (-8.72 - 6.33i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-0.896 + 1.38i)T + (-29.6 - 66.6i)T^{2} \) |
| 79 | \( 1 + (4.24 + 9.53i)T + (-52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (0.978 - 6.17i)T + (-78.9 - 25.6i)T^{2} \) |
| 89 | \( 1 + (-10.5 - 11.6i)T + (-9.30 + 88.5i)T^{2} \) |
| 97 | \( 1 + (2.72 + 17.1i)T + (-92.2 + 29.9i)T^{2} \) |
| 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
−12.64335573193334021234865419090, −11.97435371221793579220665047369, −10.57492757490662908961872340463, −9.546783522798656373345761275108, −8.587319989341199460029688651157, −7.15851068197433543653260358034, −5.88886730756046023360283201285, −5.13309211378218191923573553454, −3.74816270273198031965414829110, −1.39581107314121545323264695282,
2.39876313751571396058093917783, 4.38026632458168829001970546952, 5.14593767640431741675987249336, 6.06909541362419225635690980822, 7.73777245847085111026012866771, 9.141274687307035200843225384365, 9.742641477478632480294266508654, 11.03490433071029955162187331960, 11.80636030565306054729198371560, 13.13535052281042856613941264830
