| L(s) = 1 | − 3-s + 5-s + 9-s − 2·11-s − 15-s + 6·17-s − 8·19-s + 23-s + 25-s − 27-s − 8·29-s + 8·31-s + 2·33-s − 2·37-s + 6·41-s − 6·43-s + 45-s + 4·47-s − 7·49-s − 6·51-s − 6·53-s − 2·55-s + 8·57-s + 2·61-s + 2·67-s − 69-s − 14·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1/3·9-s − 0.603·11-s − 0.258·15-s + 1.45·17-s − 1.83·19-s + 0.208·23-s + 1/5·25-s − 0.192·27-s − 1.48·29-s + 1.43·31-s + 0.348·33-s − 0.328·37-s + 0.937·41-s − 0.914·43-s + 0.149·45-s + 0.583·47-s − 49-s − 0.840·51-s − 0.824·53-s − 0.269·55-s + 1.05·57-s + 0.256·61-s + 0.244·67-s − 0.120·69-s − 1.66·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−7.82622716382911549206654325432, −7.01012559435243204222850882643, −6.22826917828021723957146676317, −5.71709409498521808866809160435, −4.97539618653096535185389596982, −4.22518823281127641171490375573, −3.24010591473929538093582732348, −2.28430625186162076673539427720, −1.31659011669631081353896490494, 0,
1.31659011669631081353896490494, 2.28430625186162076673539427720, 3.24010591473929538093582732348, 4.22518823281127641171490375573, 4.97539618653096535185389596982, 5.71709409498521808866809160435, 6.22826917828021723957146676317, 7.01012559435243204222850882643, 7.82622716382911549206654325432
