| L(s) = 1 | − 5-s − 7-s − 2·13-s + 6·17-s + 4·19-s + 25-s + 6·29-s + 8·31-s + 35-s − 10·37-s − 6·41-s − 8·43-s + 49-s − 6·53-s + 6·59-s + 4·61-s + 2·65-s + 14·67-s − 2·73-s + 10·79-s + 6·83-s − 6·85-s − 18·89-s + 2·91-s − 4·95-s + 8·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.377·7-s − 0.554·13-s + 1.45·17-s + 0.917·19-s + 1/5·25-s + 1.11·29-s + 1.43·31-s + 0.169·35-s − 1.64·37-s − 0.937·41-s − 1.21·43-s + 1/7·49-s − 0.824·53-s + 0.781·59-s + 0.512·61-s + 0.248·65-s + 1.71·67-s − 0.234·73-s + 1.12·79-s + 0.658·83-s − 0.650·85-s − 1.90·89-s + 0.209·91-s − 0.410·95-s + 0.812·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 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 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 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
−13.71362594865835, −13.04644643773566, −12.44694444072146, −12.09655608521675, −11.82506453184778, −11.33339033726840, −10.54467453155935, −10.14419671605855, −9.836209907117757, −9.377045374366734, −8.588086937713215, −8.180952212159193, −7.874766150954527, −7.115906549395168, −6.781912764940600, −6.326990696260380, −5.412972055633573, −5.212433217756466, −4.663039618265552, −3.870689103982777, −3.346333351195137, −3.006560207697935, −2.291006128661340, −1.395469311661712, −0.8570655487659926, 0,
0.8570655487659926, 1.395469311661712, 2.291006128661340, 3.006560207697935, 3.346333351195137, 3.870689103982777, 4.663039618265552, 5.212433217756466, 5.412972055633573, 6.326990696260380, 6.781912764940600, 7.115906549395168, 7.874766150954527, 8.180952212159193, 8.588086937713215, 9.377045374366734, 9.836209907117757, 10.14419671605855, 10.54467453155935, 11.33339033726840, 11.82506453184778, 12.09655608521675, 12.44694444072146, 13.04644643773566, 13.71362594865835
