| L(s) = 1 | − 2·5-s + 4·11-s − 13-s − 17-s + 4·19-s − 25-s − 2·29-s − 8·31-s + 2·37-s − 2·41-s + 4·43-s − 8·47-s − 7·49-s − 10·53-s − 8·55-s + 4·59-s − 14·61-s + 2·65-s + 4·67-s − 14·73-s − 8·79-s − 4·83-s + 2·85-s + 6·89-s − 8·95-s − 6·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1.20·11-s − 0.277·13-s − 0.242·17-s + 0.917·19-s − 1/5·25-s − 0.371·29-s − 1.43·31-s + 0.328·37-s − 0.312·41-s + 0.609·43-s − 1.16·47-s − 49-s − 1.37·53-s − 1.07·55-s + 0.520·59-s − 1.79·61-s + 0.248·65-s + 0.488·67-s − 1.63·73-s − 0.900·79-s − 0.439·83-s + 0.216·85-s + 0.635·89-s − 0.820·95-s − 0.609·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 127296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 127296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8090075530\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8090075530\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−13.43713563256894, −12.99721261207346, −12.48376563423399, −11.91961575193228, −11.69171888277878, −11.04324012708031, −10.94646535819905, −9.887834831594222, −9.680994942107313, −9.095986325279788, −8.680603268462558, −8.013292094390448, −7.525459647806031, −7.232133818102772, −6.549232252282178, −6.086172068659097, −5.456255454454300, −4.797565249231709, −4.327803268205541, −3.712328480887103, −3.355955832233928, −2.689389942449642, −1.689672098247563, −1.362382511101892, −0.2743457357220000,
0.2743457357220000, 1.362382511101892, 1.689672098247563, 2.689389942449642, 3.355955832233928, 3.712328480887103, 4.327803268205541, 4.797565249231709, 5.456255454454300, 6.086172068659097, 6.549232252282178, 7.232133818102772, 7.525459647806031, 8.013292094390448, 8.680603268462558, 9.095986325279788, 9.680994942107313, 9.887834831594222, 10.94646535819905, 11.04324012708031, 11.69171888277878, 11.91961575193228, 12.48376563423399, 12.99721261207346, 13.43713563256894
