L(s) = 1 | + 3-s − 5-s + 3·7-s − 2·9-s − 5·13-s − 15-s − 3·17-s + 19-s + 3·21-s + 7·23-s + 25-s − 5·27-s + 29-s + 2·31-s − 3·35-s − 2·37-s − 5·39-s − 10·41-s + 6·43-s + 2·45-s + 8·47-s + 2·49-s − 3·51-s − 9·53-s + 57-s − 5·59-s − 4·61-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1.13·7-s − 2/3·9-s − 1.38·13-s − 0.258·15-s − 0.727·17-s + 0.229·19-s + 0.654·21-s + 1.45·23-s + 1/5·25-s − 0.962·27-s + 0.185·29-s + 0.359·31-s − 0.507·35-s − 0.328·37-s − 0.800·39-s − 1.56·41-s + 0.914·43-s + 0.298·45-s + 1.16·47-s + 2/7·49-s − 0.420·51-s − 1.23·53-s + 0.132·57-s − 0.650·59-s − 0.512·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 13 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 18 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.64754828690738031878730277844, −7.32103674047455755292357769219, −6.40866698888712532126911121020, −5.27647909838592918249759305503, −4.88809774471546246399252783679, −4.10622921369590355664734711516, −3.02150681403980172134722054388, −2.48468059649855058783767421873, −1.43949216133368788524626274844, 0,
1.43949216133368788524626274844, 2.48468059649855058783767421873, 3.02150681403980172134722054388, 4.10622921369590355664734711516, 4.88809774471546246399252783679, 5.27647909838592918249759305503, 6.40866698888712532126911121020, 7.32103674047455755292357769219, 7.64754828690738031878730277844