| L(s) = 1 | − 3-s + 4·5-s + 9-s − 4·11-s + 13-s − 4·15-s − 17-s + 19-s + 2·23-s + 11·25-s − 27-s − 6·29-s + 9·31-s + 4·33-s + 11·37-s − 39-s − 10·41-s + 7·43-s + 4·45-s − 6·47-s + 51-s + 6·53-s − 16·55-s − 57-s + 8·59-s + 6·61-s + 4·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.78·5-s + 1/3·9-s − 1.20·11-s + 0.277·13-s − 1.03·15-s − 0.242·17-s + 0.229·19-s + 0.417·23-s + 11/5·25-s − 0.192·27-s − 1.11·29-s + 1.61·31-s + 0.696·33-s + 1.80·37-s − 0.160·39-s − 1.56·41-s + 1.06·43-s + 0.596·45-s − 0.875·47-s + 0.140·51-s + 0.824·53-s − 2.15·55-s − 0.132·57-s + 1.04·59-s + 0.768·61-s + 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 159936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 159936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.447466744\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.447466744\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.22272550258784, −12.99557587784001, −12.52753479990945, −11.75203824624899, −11.31130016978195, −10.87353194284035, −10.28741624815880, −9.963490098503602, −9.699133231731558, −9.012341629867414, −8.554103695047155, −7.963632092961905, −7.330894997884901, −6.829822266324418, −6.214952299468939, −5.913428478024616, −5.464128390313904, −4.875391061204422, −4.620931845603843, −3.626840115200506, −2.962027429281695, −2.315940328828496, −2.029761893834458, −1.126124478091395, −0.6088048816840150,
0.6088048816840150, 1.126124478091395, 2.029761893834458, 2.315940328828496, 2.962027429281695, 3.626840115200506, 4.620931845603843, 4.875391061204422, 5.464128390313904, 5.913428478024616, 6.214952299468939, 6.829822266324418, 7.330894997884901, 7.963632092961905, 8.554103695047155, 9.012341629867414, 9.699133231731558, 9.963490098503602, 10.28741624815880, 10.87353194284035, 11.31130016978195, 11.75203824624899, 12.52753479990945, 12.99557587784001, 13.22272550258784
