L(s) = 1 | − 3·9-s + 4·11-s − 2·13-s + 2·17-s − 4·19-s − 4·23-s − 2·29-s + 8·31-s − 6·37-s + 6·41-s + 8·43-s + 4·47-s − 6·53-s + 4·59-s + 2·61-s − 8·67-s − 6·73-s + 9·81-s − 16·83-s + 6·89-s − 14·97-s − 12·99-s − 6·101-s + 4·103-s + 14·109-s − 18·113-s + 6·117-s + ⋯ |
L(s) = 1 | − 9-s + 1.20·11-s − 0.554·13-s + 0.485·17-s − 0.917·19-s − 0.834·23-s − 0.371·29-s + 1.43·31-s − 0.986·37-s + 0.937·41-s + 1.21·43-s + 0.583·47-s − 0.824·53-s + 0.520·59-s + 0.256·61-s − 0.977·67-s − 0.702·73-s + 81-s − 1.75·83-s + 0.635·89-s − 1.42·97-s − 1.20·99-s − 0.597·101-s + 0.394·103-s + 1.34·109-s − 1.69·113-s + 0.554·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 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 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−7.36194372304734310007112788348, −6.52018592068616029985399260944, −6.04179762708888671493899602085, −5.38867194201806008960357176949, −4.41975332504676707087049724630, −3.91079757024853464020748642387, −2.95368945212330983062576397170, −2.25198268282814386124051356398, −1.20145405610619599855932818805, 0,
1.20145405610619599855932818805, 2.25198268282814386124051356398, 2.95368945212330983062576397170, 3.91079757024853464020748642387, 4.41975332504676707087049724630, 5.38867194201806008960357176949, 6.04179762708888671493899602085, 6.52018592068616029985399260944, 7.36194372304734310007112788348