L(s) = 1 | + 2·3-s − 2·5-s + 7-s + 9-s − 5·11-s − 13-s − 4·15-s − 7·17-s + 2·21-s − 23-s − 25-s − 4·27-s + 5·29-s − 3·31-s − 10·33-s − 2·35-s − 37-s − 2·39-s + 3·41-s − 4·43-s − 2·45-s + 8·47-s + 49-s − 14·51-s + 6·53-s + 10·55-s + 7·59-s + ⋯ |
L(s) = 1 | + 1.15·3-s − 0.894·5-s + 0.377·7-s + 1/3·9-s − 1.50·11-s − 0.277·13-s − 1.03·15-s − 1.69·17-s + 0.436·21-s − 0.208·23-s − 1/5·25-s − 0.769·27-s + 0.928·29-s − 0.538·31-s − 1.74·33-s − 0.338·35-s − 0.164·37-s − 0.320·39-s + 0.468·41-s − 0.609·43-s − 0.298·45-s + 1.16·47-s + 1/7·49-s − 1.96·51-s + 0.824·53-s + 1.34·55-s + 0.911·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.387414311\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.387414311\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−15.81378931927327, −15.16745276508797, −15.09484398498085, −14.14387307692264, −13.81860017915858, −13.15679430312360, −12.78199061810411, −11.98218035156882, −11.38796644981885, −10.90490974081522, −10.25319036708967, −9.642869213557354, −8.711310219679875, −8.568612037233190, −7.963767370787427, −7.400957662525971, −6.957145263778986, −5.900592400214907, −5.196600268717392, −4.419501128398939, −3.969042883106258, −3.074526321336766, −2.478010947011531, −1.958713327725291, −0.4384132467255143,
0.4384132467255143, 1.958713327725291, 2.478010947011531, 3.074526321336766, 3.969042883106258, 4.419501128398939, 5.196600268717392, 5.900592400214907, 6.957145263778986, 7.400957662525971, 7.963767370787427, 8.568612037233190, 8.711310219679875, 9.642869213557354, 10.25319036708967, 10.90490974081522, 11.38796644981885, 11.98218035156882, 12.78199061810411, 13.15679430312360, 13.81860017915858, 14.14387307692264, 15.09484398498085, 15.16745276508797, 15.81378931927327