| L(s) = 1 | − 3·5-s − 7-s + 2·13-s − 3·17-s + 4·19-s + 6·23-s + 4·25-s + 10·31-s + 3·35-s − 7·37-s − 9·41-s − 5·43-s + 3·47-s + 49-s − 6·53-s − 9·59-s + 8·61-s − 6·65-s − 8·67-s − 12·71-s − 10·73-s − 5·79-s + 9·83-s + 9·85-s − 18·89-s − 2·91-s − 12·95-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 0.377·7-s + 0.554·13-s − 0.727·17-s + 0.917·19-s + 1.25·23-s + 4/5·25-s + 1.79·31-s + 0.507·35-s − 1.15·37-s − 1.40·41-s − 0.762·43-s + 0.437·47-s + 1/7·49-s − 0.824·53-s − 1.17·59-s + 1.02·61-s − 0.744·65-s − 0.977·67-s − 1.42·71-s − 1.17·73-s − 0.562·79-s + 0.987·83-s + 0.976·85-s − 1.90·89-s − 0.209·91-s − 1.23·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−8.465158325241272081442498758497, −7.54041774906904035067206068686, −6.96128420356547084793065922737, −6.22183000747127755084060931280, −5.08209533841974709774625551123, −4.41226438166707966489183054355, −3.47318181554288494346259569891, −2.91131921985618026590043699974, −1.30438375248072222931435397848, 0,
1.30438375248072222931435397848, 2.91131921985618026590043699974, 3.47318181554288494346259569891, 4.41226438166707966489183054355, 5.08209533841974709774625551123, 6.22183000747127755084060931280, 6.96128420356547084793065922737, 7.54041774906904035067206068686, 8.465158325241272081442498758497
