| L(s) = 1 | + 5-s + 2·7-s − 11-s − 4·13-s + 2·17-s − 23-s − 4·25-s − 7·31-s + 2·35-s − 3·37-s + 8·41-s − 6·43-s + 8·47-s − 3·49-s − 6·53-s − 55-s − 5·59-s − 12·61-s − 4·65-s − 7·67-s − 3·71-s + 4·73-s − 2·77-s + 10·79-s + 6·83-s + 2·85-s − 15·89-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.755·7-s − 0.301·11-s − 1.10·13-s + 0.485·17-s − 0.208·23-s − 4/5·25-s − 1.25·31-s + 0.338·35-s − 0.493·37-s + 1.24·41-s − 0.914·43-s + 1.16·47-s − 3/7·49-s − 0.824·53-s − 0.134·55-s − 0.650·59-s − 1.53·61-s − 0.496·65-s − 0.855·67-s − 0.356·71-s + 0.468·73-s − 0.227·77-s + 1.12·79-s + 0.658·83-s + 0.216·85-s − 1.58·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−7.69242913833772582185010686143, −7.14892368575311887908494718560, −6.14904891822279137731207097129, −5.49252319411018247940447627282, −4.89830792778500137217036432202, −4.12475143940327773662034884992, −3.10894936791928072023408414397, −2.20758810273071278586475611416, −1.48884771837158153348689187504, 0,
1.48884771837158153348689187504, 2.20758810273071278586475611416, 3.10894936791928072023408414397, 4.12475143940327773662034884992, 4.89830792778500137217036432202, 5.49252319411018247940447627282, 6.14904891822279137731207097129, 7.14892368575311887908494718560, 7.69242913833772582185010686143
