| L(s) = 1 | + (0.0713 − 0.997i)2-s + (0.977 − 0.212i)3-s + (−0.989 − 0.142i)4-s + (1.58 − 1.58i)5-s + (−0.142 − 0.989i)6-s + (−1.99 + 1.09i)7-s + (−0.212 + 0.977i)8-s + (0.909 − 0.415i)9-s + (−1.46 − 1.68i)10-s + (−1.01 − 0.879i)11-s + (−0.997 + 0.0713i)12-s + (2.27 − 4.15i)13-s + (0.945 + 2.06i)14-s + (1.20 − 1.88i)15-s + (0.959 + 0.281i)16-s + (0.997 + 0.746i)17-s + ⋯ |
| L(s) = 1 | + (0.0504 − 0.705i)2-s + (0.564 − 0.122i)3-s + (−0.494 − 0.0711i)4-s + (0.706 − 0.707i)5-s + (−0.0580 − 0.404i)6-s + (−0.754 + 0.412i)7-s + (−0.0751 + 0.345i)8-s + (0.303 − 0.138i)9-s + (−0.463 − 0.534i)10-s + (−0.306 − 0.265i)11-s + (−0.287 + 0.0205i)12-s + (0.629 − 1.15i)13-s + (0.252 + 0.553i)14-s + (0.311 − 0.485i)15-s + (0.239 + 0.0704i)16-s + (0.242 + 0.181i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.481 + 0.876i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.481 + 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.896137 - 1.51510i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.896137 - 1.51510i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.0713 + 0.997i)T \) |
| 3 | \( 1 + (-0.977 + 0.212i)T \) |
| 5 | \( 1 + (-1.58 + 1.58i)T \) |
| 23 | \( 1 + (4.58 + 1.40i)T \) |
| good | 7 | \( 1 + (1.99 - 1.09i)T + (3.78 - 5.88i)T^{2} \) |
| 11 | \( 1 + (1.01 + 0.879i)T + (1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-2.27 + 4.15i)T + (-7.02 - 10.9i)T^{2} \) |
| 17 | \( 1 + (-0.997 - 0.746i)T + (4.78 + 16.3i)T^{2} \) |
| 19 | \( 1 + (-1.03 + 7.18i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (-3.31 + 0.476i)T + (27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (1.63 - 1.05i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (-7.07 - 2.63i)T + (27.9 + 24.2i)T^{2} \) |
| 41 | \( 1 + (1.59 - 3.49i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (-0.565 - 2.60i)T + (-39.1 + 17.8i)T^{2} \) |
| 47 | \( 1 + (1.64 + 1.64i)T + 47iT^{2} \) |
| 53 | \( 1 + (-3.39 - 6.22i)T + (-28.6 + 44.5i)T^{2} \) |
| 59 | \( 1 + (2.06 + 7.01i)T + (-49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (-4.62 - 7.19i)T + (-25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (0.541 + 0.0387i)T + (66.3 + 9.53i)T^{2} \) |
| 71 | \( 1 + (3.82 + 4.41i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (-1.74 - 2.33i)T + (-20.5 + 70.0i)T^{2} \) |
| 79 | \( 1 + (-4.55 + 1.33i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (2.64 - 7.07i)T + (-62.7 - 54.3i)T^{2} \) |
| 89 | \( 1 + (1.22 + 0.789i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (-5.40 - 14.4i)T + (-73.3 + 63.5i)T^{2} \) |
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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
−10.04561631993530783694326788087, −9.413555421952858003956907242994, −8.638413164053362517959230289613, −7.938389086997618145611067174816, −6.43136496527140189349777915085, −5.61206678259623301694035966672, −4.56219649742492181127794910191, −3.20595984863607785870208183056, −2.45259740092615849540309109604, −0.893249385128017996360112756578,
1.90156267358328000984984740242, 3.35344195784303761943396073477, 4.14881943425263247939300727500, 5.63935226878507790205548189743, 6.38327320704369202958594011320, 7.18175277473127611666788881728, 8.027607548545669254609879547397, 9.093974607007278656319331117882, 9.890977398022692501954304683380, 10.28514138725912467075641580109
