| 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
−11.59636253172609138218444329372, −10.45399485105972840501162676876, −10.12793670507697651500615688467, −9.164705649912392182075288498651, −7.83301605818607227873453112897, −6.38856156156336152102120630023, −5.58594004771528133263168729847, −4.27886735689465097654365430104, −3.63526361059427802784469437304, −1.74985459986340140922316658267,
1.65878155497022261172973501041, 3.24912314342336251818195180462, 4.64254351575145519791065643664, 6.03866204977753312341800343644, 6.60623695570255131060853844528, 7.42997195159839132241725989582, 8.794233191337528405722610832833, 9.527456773943441343116643726340, 11.10144403133877163548622293344, 11.71088165842121956598063966625
