| 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 | \( 1 + (2.72 - 17.1i)T + (-92.2 - 29.9i)T^{2} \) |
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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
−13.13535052281042856613941264830, −11.80636030565306054729198371560, −11.03490433071029955162187331960, −9.742641477478632480294266508654, −9.141274687307035200843225384365, −7.73777245847085111026012866771, −6.06909541362419225635690980822, −5.14593767640431741675987249336, −4.38026632458168829001970546952, −2.39876313751571396058093917783,
1.39581107314121545323264695282, 3.74816270273198031965414829110, 5.13309211378218191923573553454, 5.88886730756046023360283201285, 7.15851068197433543653260358034, 8.587319989341199460029688651157, 9.546783522798656373345761275108, 10.57492757490662908961872340463, 11.97435371221793579220665047369, 12.64335573193334021234865419090
