L(s) = 1 | + (−0.253 − 0.183i)2-s + (−0.278 − 0.857i)4-s + (1.04 + 1.44i)5-s + (−0.183 + 0.565i)8-s − 0.557i·10-s + (0.987 + 0.156i)11-s + (−0.587 + 0.809i)13-s + (−0.579 + 0.420i)16-s + (0.944 − 1.30i)20-s + (−0.221 − 0.221i)22-s + (−0.672 + 2.06i)25-s + (0.297 − 0.0966i)26-s + 0.819·32-s + (−1.00 + 0.327i)40-s + (−0.437 + 1.34i)41-s + ⋯ |
L(s) = 1 | + (−0.253 − 0.183i)2-s + (−0.278 − 0.857i)4-s + (1.04 + 1.44i)5-s + (−0.183 + 0.565i)8-s − 0.557i·10-s + (0.987 + 0.156i)11-s + (−0.587 + 0.809i)13-s + (−0.579 + 0.420i)16-s + (0.944 − 1.30i)20-s + (−0.221 − 0.221i)22-s + (−0.672 + 2.06i)25-s + (0.297 − 0.0966i)26-s + 0.819·32-s + (−1.00 + 0.327i)40-s + (−0.437 + 1.34i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 - 0.390i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 - 0.390i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.047859514\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.047859514\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 + (-0.987 - 0.156i)T \) |
| 13 | \( 1 + (0.587 - 0.809i)T \) |
good | 2 | \( 1 + (0.253 + 0.183i)T + (0.309 + 0.951i)T^{2} \) |
| 5 | \( 1 + (-1.04 - 1.44i)T + (-0.309 + 0.951i)T^{2} \) |
| 7 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 17 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 19 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 31 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 37 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 41 | \( 1 + (0.437 - 1.34i)T + (-0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 + 1.90iT - T^{2} \) |
| 47 | \( 1 + (0.297 + 0.0966i)T + (0.809 + 0.587i)T^{2} \) |
| 53 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 59 | \( 1 + (-1.87 + 0.610i)T + (0.809 - 0.587i)T^{2} \) |
| 61 | \( 1 + (0.951 + 1.30i)T + (-0.309 + 0.951i)T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (-0.831 - 1.14i)T + (-0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 79 | \( 1 + (-0.951 + 1.30i)T + (-0.309 - 0.951i)T^{2} \) |
| 83 | \( 1 + (1.44 - 1.04i)T + (0.309 - 0.951i)T^{2} \) |
| 89 | \( 1 + 1.41iT - T^{2} \) |
| 97 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
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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
−9.848351778022974167053105305424, −9.464144938259570927967227856497, −8.569066697589940632443951638609, −7.10190864172118435789678059727, −6.64338391135736148430127263426, −5.89999298233934493654339669453, −4.97845872323232474986073167638, −3.74346272672904488739598628874, −2.43438130357248464890607657467, −1.68494239966504095853524412697,
1.10112287308078184625269856681, 2.53494176274720712252459543724, 3.85943584651810003672582783898, 4.74328532599388483840580091704, 5.57415748342573500685568134024, 6.49135045898740852722632494737, 7.54692986163057407825656142747, 8.383264612541412580545843415468, 8.993758052518236341858405799370, 9.525511698360851773178937762978