| L(s) = 1 | − 3-s + 5-s − 1.84·7-s + 9-s − 0.832·11-s + 3.21·13-s − 15-s − 7.95·17-s − 4.20·19-s + 1.84·21-s + 6.57·23-s + 25-s − 27-s + 3.03·29-s + 4.73·31-s + 0.832·33-s − 1.84·35-s − 37-s − 3.21·39-s + 2.73·41-s + 5.03·43-s + 45-s − 9.11·47-s − 3.59·49-s + 7.95·51-s + 3.95·53-s − 0.832·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.697·7-s + 0.333·9-s − 0.251·11-s + 0.892·13-s − 0.258·15-s − 1.92·17-s − 0.965·19-s + 0.402·21-s + 1.37·23-s + 0.200·25-s − 0.192·27-s + 0.564·29-s + 0.849·31-s + 0.144·33-s − 0.311·35-s − 0.164·37-s − 0.515·39-s + 0.426·41-s + 0.768·43-s + 0.149·45-s − 1.32·47-s − 0.513·49-s + 1.11·51-s + 0.542·53-s − 0.112·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 37 | \( 1 + T \) |
| good | 7 | \( 1 + 1.84T + 7T^{2} \) |
| 11 | \( 1 + 0.832T + 11T^{2} \) |
| 13 | \( 1 - 3.21T + 13T^{2} \) |
| 17 | \( 1 + 7.95T + 17T^{2} \) |
| 19 | \( 1 + 4.20T + 19T^{2} \) |
| 23 | \( 1 - 6.57T + 23T^{2} \) |
| 29 | \( 1 - 3.03T + 29T^{2} \) |
| 31 | \( 1 - 4.73T + 31T^{2} \) |
| 41 | \( 1 - 2.73T + 41T^{2} \) |
| 43 | \( 1 - 5.03T + 43T^{2} \) |
| 47 | \( 1 + 9.11T + 47T^{2} \) |
| 53 | \( 1 - 3.95T + 53T^{2} \) |
| 59 | \( 1 - 3.69T + 59T^{2} \) |
| 61 | \( 1 - 5.40T + 61T^{2} \) |
| 67 | \( 1 + 11.7T + 67T^{2} \) |
| 71 | \( 1 - 13.7T + 71T^{2} \) |
| 73 | \( 1 + 3.52T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 3.55T + 83T^{2} \) |
| 89 | \( 1 + 6.55T + 89T^{2} \) |
| 97 | \( 1 + 10.4T + 97T^{2} \) |
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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
−7.11790071791806758192821372123, −6.44280857782024989072411230992, −6.36450621300359236933877196095, −5.35805040781080493145770898447, −4.63598301410855359800612545627, −4.00062950984628473068647801818, −2.96083576222378803795965722755, −2.23776381294410651939168593090, −1.13505882663623677034149113260, 0,
1.13505882663623677034149113260, 2.23776381294410651939168593090, 2.96083576222378803795965722755, 4.00062950984628473068647801818, 4.63598301410855359800612545627, 5.35805040781080493145770898447, 6.36450621300359236933877196095, 6.44280857782024989072411230992, 7.11790071791806758192821372123
