| L(s) = 1 | + 3·3-s − 14·5-s + 9·9-s − 4·11-s − 54·13-s − 42·15-s + 14·17-s + 92·19-s + 152·23-s + 71·25-s + 27·27-s − 106·29-s − 144·31-s − 12·33-s + 158·37-s − 162·39-s + 390·41-s + 508·43-s − 126·45-s − 528·47-s + 42·51-s + 606·53-s + 56·55-s + 276·57-s − 364·59-s − 678·61-s + 756·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.25·5-s + 1/3·9-s − 0.109·11-s − 1.15·13-s − 0.722·15-s + 0.199·17-s + 1.11·19-s + 1.37·23-s + 0.567·25-s + 0.192·27-s − 0.678·29-s − 0.834·31-s − 0.0633·33-s + 0.702·37-s − 0.665·39-s + 1.48·41-s + 1.80·43-s − 0.417·45-s − 1.63·47-s + 0.115·51-s + 1.57·53-s + 0.137·55-s + 0.641·57-s − 0.803·59-s − 1.42·61-s + 1.44·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 14 T + p^{3} T^{2} \) |
| 11 | \( 1 + 4 T + p^{3} T^{2} \) |
| 13 | \( 1 + 54 T + p^{3} T^{2} \) |
| 17 | \( 1 - 14 T + p^{3} T^{2} \) |
| 19 | \( 1 - 92 T + p^{3} T^{2} \) |
| 23 | \( 1 - 152 T + p^{3} T^{2} \) |
| 29 | \( 1 + 106 T + p^{3} T^{2} \) |
| 31 | \( 1 + 144 T + p^{3} T^{2} \) |
| 37 | \( 1 - 158 T + p^{3} T^{2} \) |
| 41 | \( 1 - 390 T + p^{3} T^{2} \) |
| 43 | \( 1 - 508 T + p^{3} T^{2} \) |
| 47 | \( 1 + 528 T + p^{3} T^{2} \) |
| 53 | \( 1 - 606 T + p^{3} T^{2} \) |
| 59 | \( 1 + 364 T + p^{3} T^{2} \) |
| 61 | \( 1 + 678 T + p^{3} T^{2} \) |
| 67 | \( 1 + 844 T + p^{3} T^{2} \) |
| 71 | \( 1 - 8 T + p^{3} T^{2} \) |
| 73 | \( 1 - 422 T + p^{3} T^{2} \) |
| 79 | \( 1 + 384 T + p^{3} T^{2} \) |
| 83 | \( 1 + 548 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1194 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1502 T + p^{3} T^{2} \) |
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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
−8.058018946435440175812882884886, −7.38994591747810355417693611661, −7.25906402230373360239863508439, −5.82877108374002325287728538338, −4.88436170583960733903220448022, −4.15382878160483946848362619754, −3.26140185137455311161483149323, −2.56032020426206513745262221318, −1.14399748812826999310452274977, 0,
1.14399748812826999310452274977, 2.56032020426206513745262221318, 3.26140185137455311161483149323, 4.15382878160483946848362619754, 4.88436170583960733903220448022, 5.82877108374002325287728538338, 7.25906402230373360239863508439, 7.38994591747810355417693611661, 8.058018946435440175812882884886
