L(s) = 1 | + 3-s − 2·5-s + 9-s + 11-s − 2·13-s − 2·15-s − 6·17-s + 4·23-s − 25-s + 27-s + 2·29-s + 33-s − 10·37-s − 2·39-s − 6·41-s − 8·43-s − 2·45-s + 4·47-s − 6·51-s − 6·53-s − 2·55-s + 12·59-s − 2·61-s + 4·65-s + 4·67-s + 4·69-s + 12·71-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.894·5-s + 1/3·9-s + 0.301·11-s − 0.554·13-s − 0.516·15-s − 1.45·17-s + 0.834·23-s − 1/5·25-s + 0.192·27-s + 0.371·29-s + 0.174·33-s − 1.64·37-s − 0.320·39-s − 0.937·41-s − 1.21·43-s − 0.298·45-s + 0.583·47-s − 0.840·51-s − 0.824·53-s − 0.269·55-s + 1.56·59-s − 0.256·61-s + 0.496·65-s + 0.488·67-s + 0.481·69-s + 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.548290530\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.548290530\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 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 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−15.96418956693212, −15.71160386681829, −15.07073396088244, −14.83231532656858, −13.83025801700200, −13.66869024907895, −12.82737948849049, −12.30225305837159, −11.73723944966493, −11.15500920249618, −10.59152789981063, −9.841516372949575, −9.230371083402422, −8.597038650238980, −8.202220359699161, −7.451601597868428, −6.809652690086707, −6.476122037438251, −5.119374981305609, −4.834152871869076, −3.826967474850850, −3.508855986237385, −2.490585048244883, −1.808614525456164, −0.5254636043949540,
0.5254636043949540, 1.808614525456164, 2.490585048244883, 3.508855986237385, 3.826967474850850, 4.834152871869076, 5.119374981305609, 6.476122037438251, 6.809652690086707, 7.451601597868428, 8.202220359699161, 8.597038650238980, 9.230371083402422, 9.841516372949575, 10.59152789981063, 11.15500920249618, 11.73723944966493, 12.30225305837159, 12.82737948849049, 13.66869024907895, 13.83025801700200, 14.83231532656858, 15.07073396088244, 15.71160386681829, 15.96418956693212