L(s) = 1 | + (0.5 + 0.866i)2-s + (0.664 + 1.15i)3-s + (−0.499 + 0.866i)4-s + (−0.664 + 1.15i)6-s + 2.32·7-s − 0.999·8-s + (0.616 − 1.06i)9-s + 6.39·11-s − 1.32·12-s + (0.429 − 0.743i)13-s + (1.16 + 2.01i)14-s + (−0.5 − 0.866i)16-s + (−2.34 − 4.06i)17-s + 1.23·18-s + (3.75 + 2.21i)19-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (0.383 + 0.664i)3-s + (−0.249 + 0.433i)4-s + (−0.271 + 0.469i)6-s + 0.880·7-s − 0.353·8-s + (0.205 − 0.355i)9-s + 1.92·11-s − 0.383·12-s + (0.119 − 0.206i)13-s + (0.311 + 0.539i)14-s + (−0.125 − 0.216i)16-s + (−0.568 − 0.985i)17-s + 0.290·18-s + (0.860 + 0.509i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.221 - 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.221 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.01216 + 1.60634i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.01216 + 1.60634i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (-3.75 - 2.21i)T \) |
good | 3 | \( 1 + (-0.664 - 1.15i)T + (-1.5 + 2.59i)T^{2} \) |
| 7 | \( 1 - 2.32T + 7T^{2} \) |
| 11 | \( 1 - 6.39T + 11T^{2} \) |
| 13 | \( 1 + (-0.429 + 0.743i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (2.34 + 4.06i)T + (-8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (1.73 - 3.00i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.21 + 3.82i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 8.25T + 31T^{2} \) |
| 37 | \( 1 + 9.76T + 37T^{2} \) |
| 41 | \( 1 + (1.84 + 3.19i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.56 - 6.17i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (3.59 - 6.22i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (3.10 - 5.37i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.09 - 5.35i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.01 + 6.94i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.45 - 4.24i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (1.10 + 1.90i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (2.32 + 4.03i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (5.79 + 10.0i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 6.07T + 83T^{2} \) |
| 89 | \( 1 + (5.64 - 9.78i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (5.67 + 9.82i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.912988399368212399420982013094, −9.234328811828112258183545514316, −8.708769626162057522615074803112, −7.60619945614018153056356869345, −6.85164382955672137152257095800, −5.88526795394336365702027705385, −4.81821699422194481113673298727, −4.05059036568680158179108688892, −3.29061187106906006373359409763, −1.47483832602227039304648821405,
1.41123166081409293411567953934, 1.96595759283898213887532636994, 3.50397597573142819098099045350, 4.36544867130133885585811870202, 5.34216392398065583932203364259, 6.61336321418382467141739162368, 7.16970606853695673957843703370, 8.482513944518071658262909427277, 8.816844303690172395916834701507, 9.942426319338394921548003957492