L(s) = 1 | − 2·3-s + 5-s + 2·7-s + 9-s + 13-s − 2·15-s − 5·17-s − 6·19-s − 4·21-s − 2·23-s − 4·25-s + 4·27-s + 9·29-s + 2·31-s + 2·35-s − 3·37-s − 2·39-s − 5·41-s + 45-s − 2·47-s − 3·49-s + 10·51-s + 9·53-s + 12·57-s − 8·59-s + 6·61-s + 2·63-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s + 0.277·13-s − 0.516·15-s − 1.21·17-s − 1.37·19-s − 0.872·21-s − 0.417·23-s − 4/5·25-s + 0.769·27-s + 1.67·29-s + 0.359·31-s + 0.338·35-s − 0.493·37-s − 0.320·39-s − 0.780·41-s + 0.149·45-s − 0.291·47-s − 3/7·49-s + 1.40·51-s + 1.23·53-s + 1.58·57-s − 1.04·59-s + 0.768·61-s + 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−8.593179580013735766728633582532, −8.258248518859278438486553214420, −6.88693062711107630940489538145, −6.37697744421228411758469042116, −5.64659770280368303942371761613, −4.78420083160118801963993430192, −4.16616987865897129931233797172, −2.57511517090571223097469756394, −1.52544397358253614737991046868, 0,
1.52544397358253614737991046868, 2.57511517090571223097469756394, 4.16616987865897129931233797172, 4.78420083160118801963993430192, 5.64659770280368303942371761613, 6.37697744421228411758469042116, 6.88693062711107630940489538145, 8.258248518859278438486553214420, 8.593179580013735766728633582532