L(s) = 1 | + 3-s + 5-s + 4·7-s + 9-s − 2·13-s + 15-s + 6·17-s − 4·19-s + 4·21-s + 25-s + 27-s + 6·29-s − 8·31-s + 4·35-s − 2·37-s − 2·39-s − 6·41-s − 4·43-s + 45-s + 9·49-s + 6·51-s + 6·53-s − 4·57-s + 10·61-s + 4·63-s − 2·65-s − 4·67-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1.51·7-s + 1/3·9-s − 0.554·13-s + 0.258·15-s + 1.45·17-s − 0.917·19-s + 0.872·21-s + 1/5·25-s + 0.192·27-s + 1.11·29-s − 1.43·31-s + 0.676·35-s − 0.328·37-s − 0.320·39-s − 0.937·41-s − 0.609·43-s + 0.149·45-s + 9/7·49-s + 0.840·51-s + 0.824·53-s − 0.529·57-s + 1.28·61-s + 0.503·63-s − 0.248·65-s − 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.457320751\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.457320751\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−10.10989512091566913391448478748, −9.082506062594783472831133921500, −8.307525687782695431948900879973, −7.71052407307967077644420749342, −6.76895238282715006434981339956, −5.48025531888955534300415720953, −4.84143328212506167706839183620, −3.69936628544405197022333195268, −2.39041205999551187286748903960, −1.43294542786364492056174526795,
1.43294542786364492056174526795, 2.39041205999551187286748903960, 3.69936628544405197022333195268, 4.84143328212506167706839183620, 5.48025531888955534300415720953, 6.76895238282715006434981339956, 7.71052407307967077644420749342, 8.307525687782695431948900879973, 9.082506062594783472831133921500, 10.10989512091566913391448478748