| L(s) = 1 | + 4·3-s + 7·7-s − 11·9-s + 68·11-s − 22·13-s + 30·17-s + 108·19-s + 28·21-s − 184·23-s − 152·27-s + 166·29-s − 32·31-s + 272·33-s + 370·37-s − 88·39-s + 154·41-s − 212·43-s + 512·47-s + 49·49-s + 120·51-s + 98·53-s + 432·57-s − 860·59-s + 390·61-s − 77·63-s − 60·67-s − 736·69-s + ⋯ |
| L(s) = 1 | + 0.769·3-s + 0.377·7-s − 0.407·9-s + 1.86·11-s − 0.469·13-s + 0.428·17-s + 1.30·19-s + 0.290·21-s − 1.66·23-s − 1.08·27-s + 1.06·29-s − 0.185·31-s + 1.43·33-s + 1.64·37-s − 0.361·39-s + 0.586·41-s − 0.751·43-s + 1.58·47-s + 1/7·49-s + 0.329·51-s + 0.253·53-s + 1.00·57-s − 1.89·59-s + 0.818·61-s − 0.153·63-s − 0.109·67-s − 1.28·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.032475865\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.032475865\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - p T \) |
| good | 3 | \( 1 - 4 T + p^{3} T^{2} \) |
| 11 | \( 1 - 68 T + p^{3} T^{2} \) |
| 13 | \( 1 + 22 T + p^{3} T^{2} \) |
| 17 | \( 1 - 30 T + p^{3} T^{2} \) |
| 19 | \( 1 - 108 T + p^{3} T^{2} \) |
| 23 | \( 1 + 8 p T + p^{3} T^{2} \) |
| 29 | \( 1 - 166 T + p^{3} T^{2} \) |
| 31 | \( 1 + 32 T + p^{3} T^{2} \) |
| 37 | \( 1 - 10 p T + p^{3} T^{2} \) |
| 41 | \( 1 - 154 T + p^{3} T^{2} \) |
| 43 | \( 1 + 212 T + p^{3} T^{2} \) |
| 47 | \( 1 - 512 T + p^{3} T^{2} \) |
| 53 | \( 1 - 98 T + p^{3} T^{2} \) |
| 59 | \( 1 + 860 T + p^{3} T^{2} \) |
| 61 | \( 1 - 390 T + p^{3} T^{2} \) |
| 67 | \( 1 + 60 T + p^{3} T^{2} \) |
| 71 | \( 1 - 840 T + p^{3} T^{2} \) |
| 73 | \( 1 - 630 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1312 T + p^{3} T^{2} \) |
| 83 | \( 1 - 436 T + p^{3} T^{2} \) |
| 89 | \( 1 + 598 T + p^{3} T^{2} \) |
| 97 | \( 1 + 914 T + p^{3} T^{2} \) |
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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
−9.634873549185378531767452984709, −9.362381610882289001276508766854, −8.271108151107865410269114846109, −7.66675875385133230701243771181, −6.53390468916797843701214004177, −5.61765268867566168370284252593, −4.31071958380060072385664451064, −3.46403627823485762052894649912, −2.29046396215079493714388983065, −1.02778231831586739688374608870,
1.02778231831586739688374608870, 2.29046396215079493714388983065, 3.46403627823485762052894649912, 4.31071958380060072385664451064, 5.61765268867566168370284252593, 6.53390468916797843701214004177, 7.66675875385133230701243771181, 8.271108151107865410269114846109, 9.362381610882289001276508766854, 9.634873549185378531767452984709
