| L(s) = 1 | + (0.262 − 0.0236i)2-s + (−0.333 + 2.46i)3-s + (−1.89 + 0.344i)4-s + (0.786 + 0.571i)5-s + (−0.0293 + 0.654i)6-s + (2.26 − 1.35i)7-s + (−0.999 + 0.275i)8-s + (−3.05 − 0.843i)9-s + (0.220 + 0.131i)10-s + (2.66 − 4.03i)11-s + (−0.215 − 4.79i)12-s + (0.412 + 0.625i)13-s + (0.562 − 0.408i)14-s + (−1.66 + 1.74i)15-s + (3.35 − 1.26i)16-s + (−2.00 + 6.16i)17-s + ⋯ |
| L(s) = 1 | + (0.185 − 0.0167i)2-s + (−0.192 + 1.42i)3-s + (−0.949 + 0.172i)4-s + (0.351 + 0.255i)5-s + (−0.0120 + 0.267i)6-s + (0.854 − 0.510i)7-s + (−0.353 + 0.0975i)8-s + (−1.01 − 0.281i)9-s + (0.0696 + 0.0415i)10-s + (0.802 − 1.21i)11-s + (−0.0621 − 1.38i)12-s + (0.114 + 0.173i)13-s + (0.150 − 0.109i)14-s + (−0.430 + 0.450i)15-s + (0.839 − 0.315i)16-s + (−0.485 + 1.49i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.378 - 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.378 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.736882 + 0.495066i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.736882 + 0.495066i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 71 | \( 1 + (0.0401 - 8.42i)T \) |
| good | 2 | \( 1 + (-0.262 + 0.0236i)T + (1.96 - 0.357i)T^{2} \) |
| 3 | \( 1 + (0.333 - 2.46i)T + (-2.89 - 0.798i)T^{2} \) |
| 5 | \( 1 + (-0.786 - 0.571i)T + (1.54 + 4.75i)T^{2} \) |
| 7 | \( 1 + (-2.26 + 1.35i)T + (3.31 - 6.16i)T^{2} \) |
| 11 | \( 1 + (-2.66 + 4.03i)T + (-4.32 - 10.1i)T^{2} \) |
| 13 | \( 1 + (-0.412 - 0.625i)T + (-5.10 + 11.9i)T^{2} \) |
| 17 | \( 1 + (2.00 - 6.16i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (4.24 + 4.44i)T + (-0.852 + 18.9i)T^{2} \) |
| 23 | \( 1 + (0.375 + 1.64i)T + (-20.7 + 9.97i)T^{2} \) |
| 29 | \( 1 + (-2.51 - 4.66i)T + (-15.9 + 24.2i)T^{2} \) |
| 31 | \( 1 + (3.71 + 1.39i)T + (23.3 + 20.3i)T^{2} \) |
| 37 | \( 1 + (-1.35 + 5.94i)T + (-33.3 - 16.0i)T^{2} \) |
| 41 | \( 1 + (2.34 - 2.94i)T + (-9.12 - 39.9i)T^{2} \) |
| 43 | \( 1 + (-5.78 + 5.05i)T + (5.77 - 42.6i)T^{2} \) |
| 47 | \( 1 + (-0.957 - 7.06i)T + (-45.3 + 12.5i)T^{2} \) |
| 53 | \( 1 + (7.43 + 1.34i)T + (49.6 + 18.6i)T^{2} \) |
| 59 | \( 1 + (0.302 + 6.74i)T + (-58.7 + 5.28i)T^{2} \) |
| 61 | \( 1 + (5.45 + 3.25i)T + (28.9 + 53.7i)T^{2} \) |
| 67 | \( 1 + (-8.10 + 1.47i)T + (62.7 - 23.5i)T^{2} \) |
| 73 | \( 1 + (4.11 - 0.370i)T + (71.8 - 13.0i)T^{2} \) |
| 79 | \( 1 + (17.0 - 4.71i)T + (67.8 - 40.5i)T^{2} \) |
| 83 | \( 1 + (0.158 + 3.54i)T + (-82.6 + 7.44i)T^{2} \) |
| 89 | \( 1 + (-11.5 - 2.10i)T + (83.3 + 31.2i)T^{2} \) |
| 97 | \( 1 + (-10.6 - 13.4i)T + (-21.5 + 94.5i)T^{2} \) |
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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
−14.58852260826507946848032476156, −14.20382306386423437845080464077, −12.87470267690244385055349649399, −11.13468801716039813962559655734, −10.56331880679499866094737288108, −9.185652371342947835301523486150, −8.420191071773887848823697315719, −6.09448675501022250443923235114, −4.63082054255193612411899346715, −3.81062729856081976134369397351,
1.72783365932276603986006976257, 4.64617538081496885813972194075, 5.97002070173948484431298133434, 7.38437564105423352044609170221, 8.612661061488715238410304399902, 9.739350175976749714019073390348, 11.65775096054503176486028004969, 12.42920432360604770298585225803, 13.34286589743632259704941986214, 14.20965285926465880396308019415
