| L(s) = 1 | + 2-s + 4-s + 5-s + 8-s + 10-s + 16-s + 2·17-s + 4·19-s + 20-s + 4·23-s + 25-s − 4·29-s − 31-s + 32-s + 2·34-s − 8·37-s + 4·38-s + 40-s + 6·41-s − 2·43-s + 4·46-s − 7·49-s + 50-s − 8·53-s − 4·58-s − 8·59-s − 62-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.353·8-s + 0.316·10-s + 1/4·16-s + 0.485·17-s + 0.917·19-s + 0.223·20-s + 0.834·23-s + 1/5·25-s − 0.742·29-s − 0.179·31-s + 0.176·32-s + 0.342·34-s − 1.31·37-s + 0.648·38-s + 0.158·40-s + 0.937·41-s − 0.304·43-s + 0.589·46-s − 49-s + 0.141·50-s − 1.09·53-s − 0.525·58-s − 1.04·59-s − 0.127·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 337590 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 337590 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 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
−12.79221698288719, −12.49513682641102, −11.92122214629044, −11.53314249752412, −10.99494109907357, −10.69341700099625, −10.09570485817785, −9.649815088015233, −9.201719610052883, −8.801807756413319, −8.081821671625561, −7.588361616788986, −7.310032991792791, −6.605969943097957, −6.306156607645536, −5.668058715660593, −5.261453545873405, −4.902171899148880, −4.341590220081819, −3.555009977694261, −3.309105502578360, −2.761843937077274, −2.053489460190809, −1.523081561390209, −0.9671458316550356, 0,
0.9671458316550356, 1.523081561390209, 2.053489460190809, 2.761843937077274, 3.309105502578360, 3.555009977694261, 4.341590220081819, 4.902171899148880, 5.261453545873405, 5.668058715660593, 6.306156607645536, 6.605969943097957, 7.310032991792791, 7.588361616788986, 8.081821671625561, 8.801807756413319, 9.201719610052883, 9.649815088015233, 10.09570485817785, 10.69341700099625, 10.99494109907357, 11.53314249752412, 11.92122214629044, 12.49513682641102, 12.79221698288719
