| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 8-s + 9-s − 10-s − 11-s − 12-s − 15-s + 16-s − 17-s − 18-s + 19-s + 20-s + 22-s + 23-s + 24-s + 25-s − 27-s + 29-s + 30-s + 31-s − 32-s + 33-s + 34-s + ⋯ |
| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 8-s + 9-s − 10-s − 11-s − 12-s − 15-s + 16-s − 17-s − 18-s + 19-s + 20-s + 22-s + 23-s + 24-s + 25-s − 27-s + 29-s + 30-s + 31-s − 32-s + 33-s + 34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8813497090\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8813497090\) |
| \(L(1)\) |
\(\approx\) |
\(0.6586567883\) |
| \(L(1)\) |
\(\approx\) |
\(0.6586567883\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 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
−29.57839054362177615761680270130, −28.90146241579721432166144458970, −28.363175116393515155123844083935, −27.004591930126912160633114800559, −26.18979084637576203337634552857, −24.90051294768413901571180085319, −24.11903663883344721378220969791, −22.70322486459973362529357261352, −21.45271006910613292608178575701, −20.69624110571608850830035480568, −19.08422532943578305907558274466, −17.94467880162630192686183988173, −17.54322690869305865976335259731, −16.32615453420953329084032134103, −15.41845958808207319591441781336, −13.517459778240856904249590448069, −12.253645042237072179489356510022, −10.91376516595342139041907943855, −10.19006145663115803462857767682, −9.0109789022738069214344486205, −7.34395757423439308332534976620, −6.235577657890078805368707658545, −5.1065294332994192896524229134, −2.51550982125220459718520110301, −0.91846236112798421712774832230,
0.91846236112798421712774832230, 2.51550982125220459718520110301, 5.1065294332994192896524229134, 6.235577657890078805368707658545, 7.34395757423439308332534976620, 9.0109789022738069214344486205, 10.19006145663115803462857767682, 10.91376516595342139041907943855, 12.253645042237072179489356510022, 13.517459778240856904249590448069, 15.41845958808207319591441781336, 16.32615453420953329084032134103, 17.54322690869305865976335259731, 17.94467880162630192686183988173, 19.08422532943578305907558274466, 20.69624110571608850830035480568, 21.45271006910613292608178575701, 22.70322486459973362529357261352, 24.11903663883344721378220969791, 24.90051294768413901571180085319, 26.18979084637576203337634552857, 27.004591930126912160633114800559, 28.363175116393515155123844083935, 28.90146241579721432166144458970, 29.57839054362177615761680270130
