| L(s) = 1 | − 7-s + 2.64·11-s + 5.29·17-s − 7.93·23-s − 7.93·29-s − 10·31-s + 11·37-s − 10.5·41-s + 43-s − 10.5·47-s + 49-s + 10.5·53-s + 10.5·59-s − 8·61-s − 3·67-s − 2.64·71-s − 10·73-s − 2.64·77-s − 11·79-s + 5.29·83-s + 10.5·89-s + 2·97-s + 15.8·101-s − 8·103-s + 15.8·107-s + 3·109-s − 2.64·113-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 0.797·11-s + 1.28·17-s − 1.65·23-s − 1.47·29-s − 1.79·31-s + 1.80·37-s − 1.65·41-s + 0.152·43-s − 1.54·47-s + 0.142·49-s + 1.45·53-s + 1.37·59-s − 1.02·61-s − 0.366·67-s − 0.313·71-s − 1.17·73-s − 0.301·77-s − 1.23·79-s + 0.580·83-s + 1.12·89-s + 0.203·97-s + 1.57·101-s − 0.788·103-s + 1.53·107-s + 0.287·109-s − 0.248·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 - 2.64T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 5.29T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 7.93T + 23T^{2} \) |
| 29 | \( 1 + 7.93T + 29T^{2} \) |
| 31 | \( 1 + 10T + 31T^{2} \) |
| 37 | \( 1 - 11T + 37T^{2} \) |
| 41 | \( 1 + 10.5T + 41T^{2} \) |
| 43 | \( 1 - T + 43T^{2} \) |
| 47 | \( 1 + 10.5T + 47T^{2} \) |
| 53 | \( 1 - 10.5T + 53T^{2} \) |
| 59 | \( 1 - 10.5T + 59T^{2} \) |
| 61 | \( 1 + 8T + 61T^{2} \) |
| 67 | \( 1 + 3T + 67T^{2} \) |
| 71 | \( 1 + 2.64T + 71T^{2} \) |
| 73 | \( 1 + 10T + 73T^{2} \) |
| 79 | \( 1 + 11T + 79T^{2} \) |
| 83 | \( 1 - 5.29T + 83T^{2} \) |
| 89 | \( 1 - 10.5T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−7.57394593727947946680112997706, −7.13000219741371933222216237726, −6.03618916637823544309374218199, −5.81616487820076713533972090578, −4.80120337544866269247187749279, −3.76155901299700067163746741018, −3.49256157607282103092925543093, −2.21664253462857908695625960050, −1.37191111072859135387070609048, 0,
1.37191111072859135387070609048, 2.21664253462857908695625960050, 3.49256157607282103092925543093, 3.76155901299700067163746741018, 4.80120337544866269247187749279, 5.81616487820076713533972090578, 6.03618916637823544309374218199, 7.13000219741371933222216237726, 7.57394593727947946680112997706
