| L(s) = 1 | − 3-s + 3·7-s − 2·9-s − 11-s + 6·13-s + 7·17-s − 5·19-s − 3·21-s − 6·23-s + 5·27-s + 5·29-s + 3·31-s + 33-s − 3·37-s − 6·39-s + 2·41-s + 4·43-s − 2·47-s + 2·49-s − 7·51-s + 53-s + 5·57-s + 10·59-s + 7·61-s − 6·63-s + 8·67-s + 6·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.13·7-s − 2/3·9-s − 0.301·11-s + 1.66·13-s + 1.69·17-s − 1.14·19-s − 0.654·21-s − 1.25·23-s + 0.962·27-s + 0.928·29-s + 0.538·31-s + 0.174·33-s − 0.493·37-s − 0.960·39-s + 0.312·41-s + 0.609·43-s − 0.291·47-s + 2/7·49-s − 0.980·51-s + 0.137·53-s + 0.662·57-s + 1.30·59-s + 0.896·61-s − 0.755·63-s + 0.977·67-s + 0.722·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.810701634\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.810701634\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−8.399087029652918258276580590959, −7.86128266318231762393345394649, −6.79595238655169612159978323858, −5.84390008919932440420406135915, −5.70228614026362736883198368961, −4.66684712305418596032985829235, −3.91309630544866912994909348295, −2.93022532729612946512821362162, −1.77053499937987232536017731548, −0.815642739186002380846558622440,
0.815642739186002380846558622440, 1.77053499937987232536017731548, 2.93022532729612946512821362162, 3.91309630544866912994909348295, 4.66684712305418596032985829235, 5.70228614026362736883198368961, 5.84390008919932440420406135915, 6.79595238655169612159978323858, 7.86128266318231762393345394649, 8.399087029652918258276580590959
