| L(s) = 1 | − 5-s + 6·13-s − 6·17-s + 25-s − 6·29-s + 8·31-s − 2·37-s − 2·41-s + 4·43-s − 7·49-s − 6·53-s − 4·59-s + 10·61-s − 6·65-s − 12·67-s + 10·73-s + 12·79-s + 6·85-s + 6·89-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.66·13-s − 1.45·17-s + 1/5·25-s − 1.11·29-s + 1.43·31-s − 0.328·37-s − 0.312·41-s + 0.609·43-s − 49-s − 0.824·53-s − 0.520·59-s + 1.28·61-s − 0.744·65-s − 1.46·67-s + 1.17·73-s + 1.35·79-s + 0.650·85-s + 0.635·89-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 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 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−15.09698278174142, −14.40645106857042, −13.79987232714401, −13.37409310657683, −13.01020237031616, −12.38426271669444, −11.66589219825422, −11.33068458208698, −10.82945130423904, −10.43524196707346, −9.592022702358161, −9.055176459765197, −8.625622262266916, −8.065301835507609, −7.598892224748455, −6.685805429278150, −6.454170069851245, −5.844824838619175, −5.019426424577641, −4.471832246542936, −3.818339377684879, −3.371808813575624, −2.512396760927891, −1.756676187389515, −0.9674707390642949, 0,
0.9674707390642949, 1.756676187389515, 2.512396760927891, 3.371808813575624, 3.818339377684879, 4.471832246542936, 5.019426424577641, 5.844824838619175, 6.454170069851245, 6.685805429278150, 7.598892224748455, 8.065301835507609, 8.625622262266916, 9.055176459765197, 9.592022702358161, 10.43524196707346, 10.82945130423904, 11.33068458208698, 11.66589219825422, 12.38426271669444, 13.01020237031616, 13.37409310657683, 13.79987232714401, 14.40645106857042, 15.09698278174142
