| L(s) = 1 | − 3-s − 3.56·5-s + 3.12·7-s + 9-s + 1.56·11-s − 3.56·13-s + 3.56·15-s − 6.68·17-s + 1.56·19-s − 3.12·21-s + 8·23-s + 7.68·25-s − 27-s − 1.12·29-s + 31-s − 1.56·33-s − 11.1·35-s + 8.24·37-s + 3.56·39-s − 12.2·41-s − 0.876·43-s − 3.56·45-s − 8.68·47-s + 2.75·49-s + 6.68·51-s + 2·53-s − 5.56·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.59·5-s + 1.18·7-s + 0.333·9-s + 0.470·11-s − 0.987·13-s + 0.919·15-s − 1.62·17-s + 0.358·19-s − 0.681·21-s + 1.66·23-s + 1.53·25-s − 0.192·27-s − 0.208·29-s + 0.179·31-s − 0.271·33-s − 1.88·35-s + 1.35·37-s + 0.570·39-s − 1.91·41-s − 0.133·43-s − 0.530·45-s − 1.26·47-s + 0.393·49-s + 0.936·51-s + 0.274·53-s − 0.749·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| good | 5 | \( 1 + 3.56T + 5T^{2} \) |
| 7 | \( 1 - 3.12T + 7T^{2} \) |
| 11 | \( 1 - 1.56T + 11T^{2} \) |
| 13 | \( 1 + 3.56T + 13T^{2} \) |
| 17 | \( 1 + 6.68T + 17T^{2} \) |
| 19 | \( 1 - 1.56T + 19T^{2} \) |
| 23 | \( 1 - 8T + 23T^{2} \) |
| 29 | \( 1 + 1.12T + 29T^{2} \) |
| 37 | \( 1 - 8.24T + 37T^{2} \) |
| 41 | \( 1 + 12.2T + 41T^{2} \) |
| 43 | \( 1 + 0.876T + 43T^{2} \) |
| 47 | \( 1 + 8.68T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 - 7.12T + 59T^{2} \) |
| 61 | \( 1 + 3.56T + 61T^{2} \) |
| 67 | \( 1 - 12.6T + 67T^{2} \) |
| 71 | \( 1 - 5.56T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 + 8.68T + 79T^{2} \) |
| 83 | \( 1 + 7.80T + 83T^{2} \) |
| 89 | \( 1 - 16.2T + 89T^{2} \) |
| 97 | \( 1 - 5.80T + 97T^{2} \) |
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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
−7.78046248766679213411117990815, −6.91698666163775817020192225604, −6.69950695468154618361518107410, −5.22394600768266213025859176334, −4.80948547768108425130844430760, −4.27449181578779560881306013535, −3.39977105467047963290362479017, −2.28733560571628013265265831642, −1.10447269433199495223355813497, 0,
1.10447269433199495223355813497, 2.28733560571628013265265831642, 3.39977105467047963290362479017, 4.27449181578779560881306013535, 4.80948547768108425130844430760, 5.22394600768266213025859176334, 6.69950695468154618361518107410, 6.91698666163775817020192225604, 7.78046248766679213411117990815
