| L(s) = 1 | − 5-s + 7-s + 11-s − 13-s − 17-s + 19-s − 23-s + 25-s + 29-s + 31-s − 35-s − 37-s + 41-s + 43-s + 47-s + 49-s − 53-s − 55-s − 59-s − 61-s + 65-s − 67-s + 71-s + 73-s + 77-s − 79-s − 83-s + ⋯ |
| L(s) = 1 | − 5-s + 7-s + 11-s − 13-s − 17-s + 19-s − 23-s + 25-s + 29-s + 31-s − 35-s − 37-s + 41-s + 43-s + 47-s + 49-s − 53-s − 55-s − 59-s − 61-s + 65-s − 67-s + 71-s + 73-s + 77-s − 79-s − 83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1068 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1068 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.378907357\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.378907357\) |
| \(L(1)\) |
\(\approx\) |
\(1.037592936\) |
| \(L(1)\) |
\(\approx\) |
\(1.037592936\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 89 | \( 1 \) |
| good | 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 | \( 1 - T \) |
| 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
−21.55090844219029528654784453822, −20.54882345408904717005136428401, −19.82926432620817913358284217576, −19.417363952307215234402638764442, −18.32444875820929241517057586038, −17.5540759534340394371082530232, −16.94724442862569173753386276056, −15.77501451140975089574306903941, −15.39324586199129465901791075835, −14.19105803503237525141982355261, −14.06592075330454358156204590681, −12.45003603708809874853314346386, −11.95325784991273390584655771303, −11.328531814643779221961375093412, −10.4495471402286869614757432480, −9.338655209613799579499553899273, −8.539407913400381241626295099967, −7.69885133982395807891951912699, −7.07934215185009627459581934006, −5.98968780931053622146424071192, −4.65361055583696693284746620245, −4.38838303397120434443057926847, −3.17701587015051650335911700677, −2.05135319452725079683725717177, −0.85457366495668451020363577401,
0.85457366495668451020363577401, 2.05135319452725079683725717177, 3.17701587015051650335911700677, 4.38838303397120434443057926847, 4.65361055583696693284746620245, 5.98968780931053622146424071192, 7.07934215185009627459581934006, 7.69885133982395807891951912699, 8.539407913400381241626295099967, 9.338655209613799579499553899273, 10.4495471402286869614757432480, 11.328531814643779221961375093412, 11.95325784991273390584655771303, 12.45003603708809874853314346386, 14.06592075330454358156204590681, 14.19105803503237525141982355261, 15.39324586199129465901791075835, 15.77501451140975089574306903941, 16.94724442862569173753386276056, 17.5540759534340394371082530232, 18.32444875820929241517057586038, 19.417363952307215234402638764442, 19.82926432620817913358284217576, 20.54882345408904717005136428401, 21.55090844219029528654784453822
