| L(s) = 1 | + (−1.33 + 0.467i)2-s + (−0.360 − 0.481i)3-s + (1.56 − 1.24i)4-s + (−2.13 − 0.649i)5-s + (0.705 + 0.473i)6-s + (−0.369 + 5.17i)7-s + (−1.50 + 2.39i)8-s + (0.743 − 2.53i)9-s + (3.15 − 0.133i)10-s + (−3.41 − 2.19i)11-s + (−1.16 − 0.301i)12-s + (0.206 + 2.88i)13-s + (−1.92 − 7.07i)14-s + (0.457 + 1.26i)15-s + (0.881 − 3.90i)16-s + (1.79 − 4.80i)17-s + ⋯ |
| L(s) = 1 | + (−0.943 + 0.330i)2-s + (−0.207 − 0.277i)3-s + (0.781 − 0.624i)4-s + (−0.956 − 0.290i)5-s + (0.288 + 0.193i)6-s + (−0.139 + 1.95i)7-s + (−0.530 + 0.847i)8-s + (0.247 − 0.843i)9-s + (0.999 − 0.0422i)10-s + (−1.03 − 0.662i)11-s + (−0.335 − 0.0871i)12-s + (0.0572 + 0.799i)13-s + (−0.514 − 1.89i)14-s + (0.118 + 0.326i)15-s + (0.220 − 0.975i)16-s + (0.434 − 1.16i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.704 + 0.709i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.704 + 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.520209 - 0.216413i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.520209 - 0.216413i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.33 - 0.467i)T \) |
| 5 | \( 1 + (2.13 + 0.649i)T \) |
| 23 | \( 1 + (4.73 + 0.764i)T \) |
| good | 3 | \( 1 + (0.360 + 0.481i)T + (-0.845 + 2.87i)T^{2} \) |
| 7 | \( 1 + (0.369 - 5.17i)T + (-6.92 - 0.996i)T^{2} \) |
| 11 | \( 1 + (3.41 + 2.19i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (-0.206 - 2.88i)T + (-12.8 + 1.85i)T^{2} \) |
| 17 | \( 1 + (-1.79 + 4.80i)T + (-12.8 - 11.1i)T^{2} \) |
| 19 | \( 1 + (-1.27 - 0.584i)T + (12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (-3.06 - 6.71i)T + (-18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (0.369 + 2.56i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (-4.79 + 2.61i)T + (20.0 - 31.1i)T^{2} \) |
| 41 | \( 1 + (-10.1 + 2.96i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (-1.87 + 1.40i)T + (12.1 - 41.2i)T^{2} \) |
| 47 | \( 1 + (-3.03 + 3.03i)T - 47iT^{2} \) |
| 53 | \( 1 + (-0.912 + 12.7i)T + (-52.4 - 7.54i)T^{2} \) |
| 59 | \( 1 + (2.66 - 3.07i)T + (-8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (1.54 + 10.7i)T + (-58.5 + 17.1i)T^{2} \) |
| 67 | \( 1 + (-1.67 + 7.71i)T + (-60.9 - 27.8i)T^{2} \) |
| 71 | \( 1 + (-4.50 + 2.89i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (-3.93 + 1.46i)T + (55.1 - 47.8i)T^{2} \) |
| 79 | \( 1 + (-7.10 + 8.19i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (-5.45 - 9.99i)T + (-44.8 + 69.8i)T^{2} \) |
| 89 | \( 1 + (-0.322 + 2.23i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (-6.87 - 3.75i)T + (52.4 + 81.6i)T^{2} \) |
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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
−9.523522108595561605554724595965, −9.162235342363721946263429932337, −8.319251494693100038650988198111, −7.65111940071719563291794068727, −6.62635550210364648961428979410, −5.79072859992076404232555506592, −5.05188986132531757505868286453, −3.29330565353063854742499611139, −2.23001638240007643455354888284, −0.48636563940980226767288905577,
0.956507553990610110782680814386, 2.67793329488463276526062318347, 3.89591493288111194908159573152, 4.47734599689495701637699008027, 6.10773893093823325874969246972, 7.33992290840263796247846040198, 7.77405623869879964328687265735, 8.074814419284865652840111182279, 9.782517851522067155400256381561, 10.40244915505060515525228149743
