| L(s) = 1 | + 3-s + 1.91·5-s − 2.97·7-s + 9-s + 2.04·11-s + 1.36·13-s + 1.91·15-s − 6.26·17-s − 2.09·19-s − 2.97·21-s − 23-s − 1.33·25-s + 27-s − 29-s + 1.34·31-s + 2.04·33-s − 5.70·35-s − 5.21·37-s + 1.36·39-s + 7.87·41-s − 1.58·43-s + 1.91·45-s + 3.90·47-s + 1.86·49-s − 6.26·51-s − 9.86·53-s + 3.91·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.856·5-s − 1.12·7-s + 0.333·9-s + 0.616·11-s + 0.379·13-s + 0.494·15-s − 1.51·17-s − 0.481·19-s − 0.649·21-s − 0.208·23-s − 0.266·25-s + 0.192·27-s − 0.185·29-s + 0.241·31-s + 0.355·33-s − 0.963·35-s − 0.856·37-s + 0.219·39-s + 1.22·41-s − 0.241·43-s + 0.285·45-s + 0.569·47-s + 0.266·49-s − 0.877·51-s − 1.35·53-s + 0.528·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 5 | \( 1 - 1.91T + 5T^{2} \) |
| 7 | \( 1 + 2.97T + 7T^{2} \) |
| 11 | \( 1 - 2.04T + 11T^{2} \) |
| 13 | \( 1 - 1.36T + 13T^{2} \) |
| 17 | \( 1 + 6.26T + 17T^{2} \) |
| 19 | \( 1 + 2.09T + 19T^{2} \) |
| 31 | \( 1 - 1.34T + 31T^{2} \) |
| 37 | \( 1 + 5.21T + 37T^{2} \) |
| 41 | \( 1 - 7.87T + 41T^{2} \) |
| 43 | \( 1 + 1.58T + 43T^{2} \) |
| 47 | \( 1 - 3.90T + 47T^{2} \) |
| 53 | \( 1 + 9.86T + 53T^{2} \) |
| 59 | \( 1 + 2.00T + 59T^{2} \) |
| 61 | \( 1 + 9.80T + 61T^{2} \) |
| 67 | \( 1 - 2.45T + 67T^{2} \) |
| 71 | \( 1 + 8.65T + 71T^{2} \) |
| 73 | \( 1 - 14.2T + 73T^{2} \) |
| 79 | \( 1 - 0.878T + 79T^{2} \) |
| 83 | \( 1 + 13.3T + 83T^{2} \) |
| 89 | \( 1 + 5.49T + 89T^{2} \) |
| 97 | \( 1 + 8.26T + 97T^{2} \) |
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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
−7.42221546719231494212893590710, −6.49621375962345729298747462390, −6.41895612018408804637962135536, −5.56066728005648700689432850079, −4.47421582299349357496932706425, −3.88511456305808724827557887831, −3.01710030994144607709982758437, −2.27339762770421284661699455292, −1.48479004553211314467115083028, 0,
1.48479004553211314467115083028, 2.27339762770421284661699455292, 3.01710030994144607709982758437, 3.88511456305808724827557887831, 4.47421582299349357496932706425, 5.56066728005648700689432850079, 6.41895612018408804637962135536, 6.49621375962345729298747462390, 7.42221546719231494212893590710
