L(s) = 1 | + (0.841 + 0.540i)2-s + (−0.207 − 1.44i)3-s + (0.415 + 0.909i)4-s + (1.07 + 0.314i)5-s + (0.606 − 1.32i)6-s + (−0.654 + 0.755i)7-s + (−0.142 + 0.989i)8-s + (0.832 − 0.244i)9-s + (0.731 + 0.843i)10-s + (2.60 − 1.67i)11-s + (1.22 − 0.789i)12-s + (2.50 + 2.88i)13-s + (−0.959 + 0.281i)14-s + (0.232 − 1.61i)15-s + (−0.654 + 0.755i)16-s + (0.979 − 2.14i)17-s + ⋯ |
L(s) = 1 | + (0.594 + 0.382i)2-s + (−0.119 − 0.834i)3-s + (0.207 + 0.454i)4-s + (0.479 + 0.140i)5-s + (0.247 − 0.542i)6-s + (−0.247 + 0.285i)7-s + (−0.0503 + 0.349i)8-s + (0.277 − 0.0814i)9-s + (0.231 + 0.266i)10-s + (0.784 − 0.504i)11-s + (0.354 − 0.227i)12-s + (0.694 + 0.801i)13-s + (−0.256 + 0.0752i)14-s + (0.0599 − 0.416i)15-s + (−0.163 + 0.188i)16-s + (0.237 − 0.520i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0158i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0158i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.93855 - 0.0154095i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.93855 - 0.0154095i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.841 - 0.540i)T \) |
| 7 | \( 1 + (0.654 - 0.755i)T \) |
| 23 | \( 1 + (-1.74 - 4.46i)T \) |
good | 3 | \( 1 + (0.207 + 1.44i)T + (-2.87 + 0.845i)T^{2} \) |
| 5 | \( 1 + (-1.07 - 0.314i)T + (4.20 + 2.70i)T^{2} \) |
| 11 | \( 1 + (-2.60 + 1.67i)T + (4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (-2.50 - 2.88i)T + (-1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.979 + 2.14i)T + (-11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (3.02 + 6.62i)T + (-12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (3.81 - 8.34i)T + (-18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (0.0903 - 0.628i)T + (-29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 + (0.205 - 0.0603i)T + (31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (-1.12 - 0.330i)T + (34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (-1.40 - 9.79i)T + (-41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 + 10.8T + 47T^{2} \) |
| 53 | \( 1 + (0.933 - 1.07i)T + (-7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (8.50 + 9.81i)T + (-8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (1.32 - 9.23i)T + (-58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (-4.09 - 2.63i)T + (27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (13.4 + 8.66i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (1.82 + 3.99i)T + (-47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (2.36 + 2.72i)T + (-11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (-7.06 + 2.07i)T + (69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (-0.659 - 4.58i)T + (-85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (-0.687 - 0.201i)T + (81.6 + 52.4i)T^{2} \) |
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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
−11.71088165842121956598063966625, −11.10144403133877163548622293344, −9.527456773943441343116643726340, −8.794233191337528405722610832833, −7.42997195159839132241725989582, −6.60623695570255131060853844528, −6.03866204977753312341800343644, −4.64254351575145519791065643664, −3.24912314342336251818195180462, −1.65878155497022261172973501041,
1.74985459986340140922316658267, 3.63526361059427802784469437304, 4.27886735689465097654365430104, 5.58594004771528133263168729847, 6.38856156156336152102120630023, 7.83301605818607227873453112897, 9.164705649912392182075288498651, 10.12793670507697651500615688467, 10.45399485105972840501162676876, 11.59636253172609138218444329372