| L(s) = 1 | − 4.73·3-s + 8.82·5-s + 28.0·7-s − 4.55·9-s + 43.9·11-s − 73.8·13-s − 41.7·15-s − 52.6·17-s − 134.·19-s − 133.·21-s − 47.1·25-s + 149.·27-s + 149.·29-s + 42.5·31-s − 208.·33-s + 247.·35-s − 46.7·37-s + 349.·39-s + 159.·41-s + 423.·43-s − 40.2·45-s + 427.·47-s + 446.·49-s + 249.·51-s − 703.·53-s + 387.·55-s + 637.·57-s + ⋯ |
| L(s) = 1 | − 0.911·3-s + 0.788·5-s + 1.51·7-s − 0.168·9-s + 1.20·11-s − 1.57·13-s − 0.719·15-s − 0.751·17-s − 1.62·19-s − 1.38·21-s − 0.377·25-s + 1.06·27-s + 0.959·29-s + 0.246·31-s − 1.09·33-s + 1.19·35-s − 0.207·37-s + 1.43·39-s + 0.609·41-s + 1.50·43-s − 0.133·45-s + 1.32·47-s + 1.30·49-s + 0.684·51-s − 1.82·53-s + 0.950·55-s + 1.48·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2116 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.880010556\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.880010556\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + 4.73T + 27T^{2} \) |
| 5 | \( 1 - 8.82T + 125T^{2} \) |
| 7 | \( 1 - 28.0T + 343T^{2} \) |
| 11 | \( 1 - 43.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 73.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 52.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 134.T + 6.85e3T^{2} \) |
| 29 | \( 1 - 149.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 42.5T + 2.97e4T^{2} \) |
| 37 | \( 1 + 46.7T + 5.06e4T^{2} \) |
| 41 | \( 1 - 159.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 423.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 427.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 703.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 168.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 436.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 100.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 537.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 317.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 907.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 158.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 444.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 539.T + 9.12e5T^{2} \) |
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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
−8.798999567459029570850917895779, −8.000688873413205001219842381643, −7.00090525480374584668982045940, −6.28438281662785667740070606516, −5.60706053710268088361771171418, −4.67365443851109549366430123573, −4.33180445891842381540392653129, −2.48450493756122934187239730663, −1.83840955048465475960573866486, −0.65385985657322530402091353097,
0.65385985657322530402091353097, 1.83840955048465475960573866486, 2.48450493756122934187239730663, 4.33180445891842381540392653129, 4.67365443851109549366430123573, 5.60706053710268088361771171418, 6.28438281662785667740070606516, 7.00090525480374584668982045940, 8.000688873413205001219842381643, 8.798999567459029570850917895779
