| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 8-s + 9-s − 10-s + 11-s − 12-s − 13-s + 15-s + 16-s − 17-s + 18-s − 19-s − 20-s + 22-s + 23-s − 24-s + 25-s − 26-s − 27-s + 29-s + 30-s − 31-s + 32-s + ⋯ |
| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 8-s + 9-s − 10-s + 11-s − 12-s − 13-s + 15-s + 16-s − 17-s + 18-s − 19-s − 20-s + 22-s + 23-s − 24-s + 25-s − 26-s − 27-s + 29-s + 30-s − 31-s + 32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.146585666\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.146585666\) |
| \(L(1)\) |
\(\approx\) |
\(1.187410411\) |
| \(L(1)\) |
\(\approx\) |
\(1.187410411\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 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
−50.98309943494790941040256869584, −49.2358942624162595582001378897, −47.29858534027669804962753511398, −46.19432742740581606201873969732, −43.93185428742714565723041221515, −42.30485758764265058072877014058, −40.7103286516539188858032460596, −39.48257874666192172268872121810, −38.36463733603136596844459180378, −35.311847458977969626587230969278, −34.11124078303733952175233551521, −32.4758875915420758078446454007, −30.72189964454757558763990410272, −29.17887966636347069628776553811, −27.36147751900128252344458972943, −24.49884755535632897308961801424, −23.16297179973657863840493856183, −21.89991370331832248627181534368, −19.61187805669003102874335093663, −16.802876475728853982349899698677, −15.11288225874376865962836973665, −12.48960334303313423775824003050, −11.16018454311952965510181826588, −6.84549171249137726783979779478, −4.47573828372868313197462848719,
4.47573828372868313197462848719, 6.84549171249137726783979779478, 11.16018454311952965510181826588, 12.48960334303313423775824003050, 15.11288225874376865962836973665, 16.802876475728853982349899698677, 19.61187805669003102874335093663, 21.89991370331832248627181534368, 23.16297179973657863840493856183, 24.49884755535632897308961801424, 27.36147751900128252344458972943, 29.17887966636347069628776553811, 30.72189964454757558763990410272, 32.4758875915420758078446454007, 34.11124078303733952175233551521, 35.311847458977969626587230969278, 38.36463733603136596844459180378, 39.48257874666192172268872121810, 40.7103286516539188858032460596, 42.30485758764265058072877014058, 43.93185428742714565723041221515, 46.19432742740581606201873969732, 47.29858534027669804962753511398, 49.2358942624162595582001378897, 50.98309943494790941040256869584
