| L(s) = 1 | + (−2.40 − 0.276i)2-s + (−1.30 − 0.947i)3-s + (3.78 + 0.879i)4-s + (−0.564 + 0.825i)5-s + (2.88 + 2.64i)6-s + (−2.42 − 2.49i)7-s + (−4.30 − 1.53i)8-s + (−0.123 − 0.380i)9-s + (1.58 − 1.83i)10-s + (2.90 + 1.60i)11-s + (−4.09 − 4.73i)12-s + (0.617 − 1.02i)13-s + (5.15 + 6.68i)14-s + (1.51 − 0.541i)15-s + (2.96 + 1.45i)16-s + (3.01 − 2.46i)17-s + ⋯ |
| L(s) = 1 | + (−1.70 − 0.195i)2-s + (−0.753 − 0.547i)3-s + (1.89 + 0.439i)4-s + (−0.252 + 0.369i)5-s + (1.17 + 1.07i)6-s + (−0.917 − 0.943i)7-s + (−1.52 − 0.542i)8-s + (−0.0412 − 0.126i)9-s + (0.502 − 0.579i)10-s + (0.874 + 0.485i)11-s + (−1.18 − 1.36i)12-s + (0.171 − 0.284i)13-s + (1.37 + 1.78i)14-s + (0.392 − 0.139i)15-s + (0.742 + 0.364i)16-s + (0.730 − 0.597i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.634 - 0.772i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.634 - 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0271685 + 0.0574632i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0271685 + 0.0574632i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.564 - 0.825i)T \) |
| 11 | \( 1 + (-2.90 - 1.60i)T \) |
| good | 2 | \( 1 + (2.40 + 0.276i)T + (1.94 + 0.452i)T^{2} \) |
| 3 | \( 1 + (1.30 + 0.947i)T + (0.927 + 2.85i)T^{2} \) |
| 7 | \( 1 + (2.42 + 2.49i)T + (-0.199 + 6.99i)T^{2} \) |
| 13 | \( 1 + (-0.617 + 1.02i)T + (-6.06 - 11.4i)T^{2} \) |
| 17 | \( 1 + (-3.01 + 2.46i)T + (3.37 - 16.6i)T^{2} \) |
| 19 | \( 1 + (6.97 + 0.398i)T + (18.8 + 2.16i)T^{2} \) |
| 23 | \( 1 + (-0.524 + 3.64i)T + (-22.0 - 6.47i)T^{2} \) |
| 29 | \( 1 + (-3.60 + 9.25i)T + (-21.3 - 19.6i)T^{2} \) |
| 31 | \( 1 + (-4.00 - 1.69i)T + (21.6 + 22.2i)T^{2} \) |
| 37 | \( 1 + (-0.533 + 6.21i)T + (-36.4 - 6.30i)T^{2} \) |
| 41 | \( 1 + (4.36 - 2.46i)T + (21.1 - 35.1i)T^{2} \) |
| 43 | \( 1 + (8.17 - 2.39i)T + (36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 + (2.24 - 11.0i)T + (-43.2 - 18.2i)T^{2} \) |
| 53 | \( 1 + (9.77 - 4.80i)T + (32.3 - 41.9i)T^{2} \) |
| 59 | \( 1 + (-4.49 - 2.53i)T + (30.4 + 50.5i)T^{2} \) |
| 61 | \( 1 + (10.4 - 1.19i)T + (59.4 - 13.8i)T^{2} \) |
| 67 | \( 1 + (4.16 - 9.12i)T + (-43.8 - 50.6i)T^{2} \) |
| 71 | \( 1 + (-2.81 - 10.7i)T + (-61.8 + 34.9i)T^{2} \) |
| 73 | \( 1 + (-4.08 - 7.73i)T + (-41.2 + 60.2i)T^{2} \) |
| 79 | \( 1 + (0.102 + 3.58i)T + (-78.8 + 4.51i)T^{2} \) |
| 83 | \( 1 + (-5.75 - 0.996i)T + (78.1 + 27.8i)T^{2} \) |
| 89 | \( 1 + (4.21 + 2.71i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (3.59 + 5.25i)T + (-35.1 + 90.3i)T^{2} \) |
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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
−10.06639880263996706480370913830, −9.421259023258380401140313531821, −8.334406282294295641430772519185, −7.44494325158725917191555072136, −6.54926235940819677726975746882, −6.38211033817507207103317343456, −4.23669286176547116049091946965, −2.82972116969669414813685260055, −1.16507138435920937175610477258, −0.07809129480099579845510418884,
1.71641667130778149513643443081, 3.42985480760841045159267966876, 5.00573286356841360647479957034, 6.22903037833202339791952054325, 6.60402263979210960385855897390, 8.142917931414713640816942679172, 8.659873034542960326473817504444, 9.426157131010483446349395876411, 10.21514846698361637229203764110, 10.88127109519507617223994029852
