| L(s) = 1 | + 5-s − 7-s + 13-s − 17-s + 3·19-s − 6·23-s + 25-s − 3·29-s − 35-s − 5·37-s − 12·41-s − 8·43-s − 12·47-s − 6·49-s − 2·53-s + 65-s + 12·67-s − 71-s + 8·73-s − 8·79-s − 83-s − 85-s − 6·89-s − 91-s + 3·95-s − 8·97-s + 101-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.377·7-s + 0.277·13-s − 0.242·17-s + 0.688·19-s − 1.25·23-s + 1/5·25-s − 0.557·29-s − 0.169·35-s − 0.821·37-s − 1.87·41-s − 1.21·43-s − 1.75·47-s − 6/7·49-s − 0.274·53-s + 0.124·65-s + 1.46·67-s − 0.118·71-s + 0.936·73-s − 0.900·79-s − 0.109·83-s − 0.108·85-s − 0.635·89-s − 0.104·91-s + 0.307·95-s − 0.812·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.437438029\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.437438029\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−14.60409168763892, −14.09511658455942, −13.65489268341840, −13.19791997403017, −12.67820553997972, −12.11873491389468, −11.47519313240256, −11.21138447910445, −10.30869727020044, −9.910290249913593, −9.640419254864521, −8.843686175372679, −8.308898334770781, −7.897453206422812, −6.990511880704977, −6.648745044865900, −6.085178329065698, −5.356516716178182, −4.988599897724440, −4.138169817625749, −3.398888351337852, −3.065401437495274, −1.890586287072609, −1.677739677192176, −0.4119463335781092,
0.4119463335781092, 1.677739677192176, 1.890586287072609, 3.065401437495274, 3.398888351337852, 4.138169817625749, 4.988599897724440, 5.356516716178182, 6.085178329065698, 6.648745044865900, 6.990511880704977, 7.897453206422812, 8.308898334770781, 8.843686175372679, 9.640419254864521, 9.910290249913593, 10.30869727020044, 11.21138447910445, 11.47519313240256, 12.11873491389468, 12.67820553997972, 13.19791997403017, 13.65489268341840, 14.09511658455942, 14.60409168763892
