| L(s) = 1 | + 7·3-s + 13·5-s + 16·7-s + 22·9-s + 45·11-s − 61·13-s + 91·15-s − 102·17-s + 68·19-s + 112·21-s + 194·23-s + 44·25-s − 35·27-s + 29·29-s + 149·31-s + 315·33-s + 208·35-s − 400·37-s − 427·39-s + 280·41-s − 263·43-s + 286·45-s + 509·47-s − 87·49-s − 714·51-s + 605·53-s + 585·55-s + ⋯ |
| L(s) = 1 | + 1.34·3-s + 1.16·5-s + 0.863·7-s + 0.814·9-s + 1.23·11-s − 1.30·13-s + 1.56·15-s − 1.45·17-s + 0.821·19-s + 1.16·21-s + 1.75·23-s + 0.351·25-s − 0.249·27-s + 0.185·29-s + 0.863·31-s + 1.66·33-s + 1.00·35-s − 1.77·37-s − 1.75·39-s + 1.06·41-s − 0.932·43-s + 0.947·45-s + 1.57·47-s − 0.253·49-s − 1.96·51-s + 1.56·53-s + 1.43·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.613675277\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.613675277\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 - p T \) |
| good | 3 | \( 1 - 7 T + p^{3} T^{2} \) |
| 5 | \( 1 - 13 T + p^{3} T^{2} \) |
| 7 | \( 1 - 16 T + p^{3} T^{2} \) |
| 11 | \( 1 - 45 T + p^{3} T^{2} \) |
| 13 | \( 1 + 61 T + p^{3} T^{2} \) |
| 17 | \( 1 + 6 p T + p^{3} T^{2} \) |
| 19 | \( 1 - 68 T + p^{3} T^{2} \) |
| 23 | \( 1 - 194 T + p^{3} T^{2} \) |
| 31 | \( 1 - 149 T + p^{3} T^{2} \) |
| 37 | \( 1 + 400 T + p^{3} T^{2} \) |
| 41 | \( 1 - 280 T + p^{3} T^{2} \) |
| 43 | \( 1 + 263 T + p^{3} T^{2} \) |
| 47 | \( 1 - 509 T + p^{3} T^{2} \) |
| 53 | \( 1 - 605 T + p^{3} T^{2} \) |
| 59 | \( 1 - 578 T + p^{3} T^{2} \) |
| 61 | \( 1 - 718 T + p^{3} T^{2} \) |
| 67 | \( 1 - 260 T + p^{3} T^{2} \) |
| 71 | \( 1 - 738 T + p^{3} T^{2} \) |
| 73 | \( 1 - 652 T + p^{3} T^{2} \) |
| 79 | \( 1 + 917 T + p^{3} T^{2} \) |
| 83 | \( 1 + 678 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1008 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1764 T + p^{3} T^{2} \) |
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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
−8.832596570036482288772933398166, −8.441245148099059164790491606875, −7.15802782080398865792076575422, −6.87073210980526279821966031611, −5.51990442159684255489799909271, −4.78486710205613312899518852311, −3.80403609329825345444364808073, −2.60636586513739307172221415762, −2.12348765222310633986712559340, −1.11625403927807672435950392778,
1.11625403927807672435950392778, 2.12348765222310633986712559340, 2.60636586513739307172221415762, 3.80403609329825345444364808073, 4.78486710205613312899518852311, 5.51990442159684255489799909271, 6.87073210980526279821966031611, 7.15802782080398865792076575422, 8.441245148099059164790491606875, 8.832596570036482288772933398166
