L(s) = 1 | + 5-s + 7-s − 2·11-s + 5·17-s − 2·19-s − 2·23-s − 4·25-s + 10·29-s + 35-s + 5·37-s + 3·41-s + 7·43-s + 3·47-s + 49-s + 6·53-s − 2·55-s − 59-s − 6·61-s − 4·67-s − 8·71-s + 10·73-s − 2·77-s + 3·79-s + 13·83-s + 5·85-s − 6·89-s − 2·95-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 0.377·7-s − 0.603·11-s + 1.21·17-s − 0.458·19-s − 0.417·23-s − 4/5·25-s + 1.85·29-s + 0.169·35-s + 0.821·37-s + 0.468·41-s + 1.06·43-s + 0.437·47-s + 1/7·49-s + 0.824·53-s − 0.269·55-s − 0.130·59-s − 0.768·61-s − 0.488·67-s − 0.949·71-s + 1.17·73-s − 0.227·77-s + 0.337·79-s + 1.42·83-s + 0.542·85-s − 0.635·89-s − 0.205·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.098978614\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.098978614\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
good | 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 - 13 T + p T^{2} \) |
| 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
−8.652516223529645480049385679840, −7.929996188736616575281339949094, −7.40503086367960443959848677662, −6.26440052330371122073947339882, −5.75816601145574782619220045152, −4.88631415809130809304589149622, −4.07834214482260500285759308302, −2.95567535531384475359756406475, −2.11979585235856857271660904210, −0.904163145672453187106914599362,
0.904163145672453187106914599362, 2.11979585235856857271660904210, 2.95567535531384475359756406475, 4.07834214482260500285759308302, 4.88631415809130809304589149622, 5.75816601145574782619220045152, 6.26440052330371122073947339882, 7.40503086367960443959848677662, 7.929996188736616575281339949094, 8.652516223529645480049385679840