| L(s) = 1 | − 5-s + 4·11-s + 2·13-s − 6·17-s + 4·19-s + 8·23-s + 25-s − 6·29-s − 10·37-s + 6·41-s + 43-s + 8·47-s − 7·49-s + 2·53-s − 4·55-s + 4·59-s − 14·61-s − 2·65-s − 4·67-s − 16·71-s + 10·73-s − 8·79-s − 16·83-s + 6·85-s − 6·89-s − 4·95-s + 2·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.20·11-s + 0.554·13-s − 1.45·17-s + 0.917·19-s + 1.66·23-s + 1/5·25-s − 1.11·29-s − 1.64·37-s + 0.937·41-s + 0.152·43-s + 1.16·47-s − 49-s + 0.274·53-s − 0.539·55-s + 0.520·59-s − 1.79·61-s − 0.248·65-s − 0.488·67-s − 1.89·71-s + 1.17·73-s − 0.900·79-s − 1.75·83-s + 0.650·85-s − 0.635·89-s − 0.410·95-s + 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−15.36015844768898, −14.95112630185176, −14.24532787316431, −13.84440591238146, −13.21943945312984, −12.77148972291480, −12.11548885776199, −11.57319045633504, −11.08024295584577, −10.82569463705986, −9.938088459690179, −9.206051289432301, −8.903755573712046, −8.546626413444490, −7.448439450066020, −7.229169248389388, −6.600860481078635, −5.970739839956301, −5.285277969180749, −4.544459962445154, −4.034016174845185, −3.381500707288656, −2.748104543313615, −1.694352453053431, −1.108638097864253, 0,
1.108638097864253, 1.694352453053431, 2.748104543313615, 3.381500707288656, 4.034016174845185, 4.544459962445154, 5.285277969180749, 5.970739839956301, 6.600860481078635, 7.229169248389388, 7.448439450066020, 8.546626413444490, 8.903755573712046, 9.206051289432301, 9.938088459690179, 10.82569463705986, 11.08024295584577, 11.57319045633504, 12.11548885776199, 12.77148972291480, 13.21943945312984, 13.84440591238146, 14.24532787316431, 14.95112630185176, 15.36015844768898
