| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s + 7-s − 8-s + 9-s − 10-s + 11-s − 12-s − 13-s − 14-s − 15-s + 16-s − 17-s − 18-s + 19-s + 20-s − 21-s − 22-s + 23-s + 24-s + 25-s + 26-s − 27-s + 28-s + ⋯ |
| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s + 7-s − 8-s + 9-s − 10-s + 11-s − 12-s − 13-s − 14-s − 15-s + 16-s − 17-s − 18-s + 19-s + 20-s − 21-s − 22-s + 23-s + 24-s + 25-s + 26-s − 27-s + 28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1061 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1061 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.039747365\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.039747365\) |
| \(L(1)\) |
\(\approx\) |
\(0.7665981894\) |
| \(L(1)\) |
\(\approx\) |
\(0.7665981894\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 1061 | \( 1 \) |
| good | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 + T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 + T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−21.62889855877504515836553476796, −20.6705095369820018882788305578, −19.96024270614645206616940093542, −18.88037259659325465023600943692, −18.12809136743389264633454186771, −17.53545225107815859644405451885, −17.07985584343742992933870264582, −16.52362556763229443029909578642, −15.30748108614688443783770135154, −14.68022275270032744054464153560, −13.600307189590879730703546389413, −12.50416980729005182060277232341, −11.653038072827929835018427239214, −11.15445738059633826040328743110, −10.295732419380476877537035559283, −9.485060159256704809362958766781, −8.91364767404786034837531284898, −7.570049816385779069572086998915, −6.94685405125205783244553864857, −6.083062924207327127242418029121, −5.27598923255182027050727902464, −4.37477718320407385217011860744, −2.66535352828790888236194312470, −1.6880902570004627896071449801, −0.96355922474226854348686069930,
0.96355922474226854348686069930, 1.6880902570004627896071449801, 2.66535352828790888236194312470, 4.37477718320407385217011860744, 5.27598923255182027050727902464, 6.083062924207327127242418029121, 6.94685405125205783244553864857, 7.570049816385779069572086998915, 8.91364767404786034837531284898, 9.485060159256704809362958766781, 10.295732419380476877537035559283, 11.15445738059633826040328743110, 11.653038072827929835018427239214, 12.50416980729005182060277232341, 13.600307189590879730703546389413, 14.68022275270032744054464153560, 15.30748108614688443783770135154, 16.52362556763229443029909578642, 17.07985584343742992933870264582, 17.53545225107815859644405451885, 18.12809136743389264633454186771, 18.88037259659325465023600943692, 19.96024270614645206616940093542, 20.6705095369820018882788305578, 21.62889855877504515836553476796
