| L(s) = 1 | − 2·5-s + 13-s − 2·17-s + 4·19-s − 25-s − 6·29-s − 2·37-s − 6·41-s + 12·43-s − 4·47-s − 7·49-s − 6·53-s − 8·59-s − 2·61-s − 2·65-s − 4·67-s − 12·71-s − 14·73-s + 8·83-s + 4·85-s + 18·89-s − 8·95-s − 6·97-s − 14·101-s − 16·103-s + 4·107-s − 2·109-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.277·13-s − 0.485·17-s + 0.917·19-s − 1/5·25-s − 1.11·29-s − 0.328·37-s − 0.937·41-s + 1.82·43-s − 0.583·47-s − 49-s − 0.824·53-s − 1.04·59-s − 0.256·61-s − 0.248·65-s − 0.488·67-s − 1.42·71-s − 1.63·73-s + 0.878·83-s + 0.433·85-s + 1.90·89-s − 0.820·95-s − 0.609·97-s − 1.39·101-s − 1.57·103-s + 0.386·107-s − 0.191·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−8.879911485018581552559778228194, −7.87497595233942443821507658391, −7.48206171863010651464390955453, −6.50109862386502065678161037272, −5.60510042774675865346276845379, −4.63629406156287515269954623878, −3.80747723245115179033242709882, −2.97565069723020365742526516584, −1.57297786911914117520864059523, 0,
1.57297786911914117520864059523, 2.97565069723020365742526516584, 3.80747723245115179033242709882, 4.63629406156287515269954623878, 5.60510042774675865346276845379, 6.50109862386502065678161037272, 7.48206171863010651464390955453, 7.87497595233942443821507658391, 8.879911485018581552559778228194
