L(s) = 1 | − 4·13-s + 4·17-s + 4·19-s − 4·23-s − 5·25-s − 2·29-s − 8·31-s − 6·37-s + 12·41-s + 4·43-s − 8·47-s − 6·53-s + 12·59-s − 4·61-s − 4·67-s + 12·71-s − 8·73-s − 16·79-s − 4·83-s + 4·89-s − 16·97-s − 8·101-s + 8·103-s − 8·107-s − 14·109-s − 2·113-s + ⋯ |
L(s) = 1 | − 1.10·13-s + 0.970·17-s + 0.917·19-s − 0.834·23-s − 25-s − 0.371·29-s − 1.43·31-s − 0.986·37-s + 1.87·41-s + 0.609·43-s − 1.16·47-s − 0.824·53-s + 1.56·59-s − 0.512·61-s − 0.488·67-s + 1.42·71-s − 0.936·73-s − 1.80·79-s − 0.439·83-s + 0.423·89-s − 1.62·97-s − 0.796·101-s + 0.788·103-s − 0.773·107-s − 1.34·109-s − 0.188·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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
−7.943292858197333536393193375367, −7.60060339775841296378065315494, −6.84676784569375503981366508419, −5.70606535228152180943165965335, −5.38208157474292441837754641952, −4.28119710078031357633531600946, −3.49723241996278064533857546694, −2.52239391682643836376906423915, −1.49987566281652477142815643518, 0,
1.49987566281652477142815643518, 2.52239391682643836376906423915, 3.49723241996278064533857546694, 4.28119710078031357633531600946, 5.38208157474292441837754641952, 5.70606535228152180943165965335, 6.84676784569375503981366508419, 7.60060339775841296378065315494, 7.943292858197333536393193375367