| L(s) = 1 | + (0.0877 − 1.17i)2-s + (0.194 − 0.132i)3-s + (0.612 + 0.0923i)4-s + (−0.560 + 2.45i)5-s + (−0.138 − 0.239i)6-s + (1.67 + 2.04i)7-s + (0.684 − 3.00i)8-s + (−1.07 + 2.74i)9-s + (2.82 + 0.872i)10-s + (0.439 − 1.12i)11-s + (0.131 − 0.0633i)12-s + (2.14 + 1.98i)13-s + (2.54 − 1.78i)14-s + (0.216 + 0.552i)15-s + (−2.27 − 0.700i)16-s + (0.0818 + 0.358i)17-s + ⋯ |
| L(s) = 1 | + (0.0620 − 0.828i)2-s + (0.112 − 0.0766i)3-s + (0.306 + 0.0461i)4-s + (−0.250 + 1.09i)5-s + (−0.0565 − 0.0978i)6-s + (0.633 + 0.773i)7-s + (0.242 − 1.06i)8-s + (−0.358 + 0.913i)9-s + (0.894 + 0.275i)10-s + (0.132 − 0.337i)11-s + (0.0379 − 0.0182i)12-s + (0.593 + 0.551i)13-s + (0.680 − 0.476i)14-s + (0.0560 + 0.142i)15-s + (−0.567 − 0.175i)16-s + (0.0198 + 0.0869i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 301 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.191i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 301 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.981 + 0.191i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.59314 - 0.154025i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.59314 - 0.154025i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-1.67 - 2.04i)T \) |
| 43 | \( 1 + (0.510 + 6.53i)T \) |
| good | 2 | \( 1 + (-0.0877 + 1.17i)T + (-1.97 - 0.298i)T^{2} \) |
| 3 | \( 1 + (-0.194 + 0.132i)T + (1.09 - 2.79i)T^{2} \) |
| 5 | \( 1 + (0.560 - 2.45i)T + (-4.50 - 2.16i)T^{2} \) |
| 11 | \( 1 + (-0.439 + 1.12i)T + (-8.06 - 7.48i)T^{2} \) |
| 13 | \( 1 + (-2.14 - 1.98i)T + (0.971 + 12.9i)T^{2} \) |
| 17 | \( 1 + (-0.0818 - 0.358i)T + (-15.3 + 7.37i)T^{2} \) |
| 19 | \( 1 + (-1.19 + 1.49i)T + (-4.22 - 18.5i)T^{2} \) |
| 23 | \( 1 + (4.72 + 5.92i)T + (-5.11 + 22.4i)T^{2} \) |
| 29 | \( 1 + (-0.259 + 3.46i)T + (-28.6 - 4.32i)T^{2} \) |
| 31 | \( 1 + (-0.236 + 3.16i)T + (-30.6 - 4.62i)T^{2} \) |
| 37 | \( 1 + 6.08T + 37T^{2} \) |
| 41 | \( 1 + (-2.94 + 1.41i)T + (25.5 - 32.0i)T^{2} \) |
| 47 | \( 1 + (-0.400 - 1.02i)T + (-34.4 + 31.9i)T^{2} \) |
| 53 | \( 1 + (-3.04 - 0.940i)T + (43.7 + 29.8i)T^{2} \) |
| 59 | \( 1 + (3.36 - 3.12i)T + (4.40 - 58.8i)T^{2} \) |
| 61 | \( 1 + (-0.885 - 11.8i)T + (-60.3 + 9.09i)T^{2} \) |
| 67 | \( 1 + (0.605 + 1.54i)T + (-49.1 + 45.5i)T^{2} \) |
| 71 | \( 1 + (-1.60 - 4.07i)T + (-52.0 + 48.2i)T^{2} \) |
| 73 | \( 1 + (-2.51 + 11.0i)T + (-65.7 - 31.6i)T^{2} \) |
| 79 | \( 1 + (-0.636 - 1.10i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.0821 - 1.09i)T + (-82.0 + 12.3i)T^{2} \) |
| 89 | \( 1 + (15.4 + 7.43i)T + (55.4 + 69.5i)T^{2} \) |
| 97 | \( 1 + (-4.75 - 5.96i)T + (-21.5 + 94.5i)T^{2} \) |
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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
−11.56132409184889092071468058680, −10.91992831350954919235489204765, −10.30656031955638888161170384638, −8.887015822142099928856794822784, −7.896382097184426752612169620159, −6.85752243806350162751434745671, −5.81285548250450270136715179213, −4.18945892716136432615180723660, −2.86743622341930231176775626766, −2.03896960498043439181508025693,
1.37929811807650162424354632843, 3.60000649286890082302005988357, 4.87035252662526839658625093362, 5.80018981129193848941534998729, 6.95841884331647317882036314178, 7.966289754626595747879784014431, 8.565350538451461968021926104207, 9.739900895116190688621835464320, 10.95412048461315023846688555722, 11.79950601934872785948458010944
