| L(s) = 1 | + 3-s − 2·5-s + 7-s + 9-s − 2·15-s + 6·17-s + 4·19-s + 21-s − 4·23-s − 25-s + 27-s + 6·29-s + 8·31-s − 2·35-s + 10·37-s + 10·41-s + 12·43-s − 2·45-s + 8·47-s + 49-s + 6·51-s + 6·53-s + 4·57-s − 4·59-s − 10·61-s + 63-s − 12·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s + 0.377·7-s + 1/3·9-s − 0.516·15-s + 1.45·17-s + 0.917·19-s + 0.218·21-s − 0.834·23-s − 1/5·25-s + 0.192·27-s + 1.11·29-s + 1.43·31-s − 0.338·35-s + 1.64·37-s + 1.56·41-s + 1.82·43-s − 0.298·45-s + 1.16·47-s + 1/7·49-s + 0.840·51-s + 0.824·53-s + 0.529·57-s − 0.520·59-s − 1.28·61-s + 0.125·63-s − 1.46·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.220062086\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.220062086\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−15.36285794550942, −14.56481279733384, −14.15008613204919, −13.84430612260735, −13.13063349813610, −12.28892480259116, −12.07108989685401, −11.66785735318537, −10.82841134168193, −10.39237496400697, −9.664585647411515, −9.308218360794886, −8.480014600428699, −7.951747596549428, −7.600272927483715, −7.267542695024334, −6.029474070169930, −5.891623338324621, −4.813415556745731, −4.251584594782273, −3.826408209501600, −2.861412834027470, −2.589146978536198, −1.283198516586057, −0.7706984736874633,
0.7706984736874633, 1.283198516586057, 2.589146978536198, 2.861412834027470, 3.826408209501600, 4.251584594782273, 4.813415556745731, 5.891623338324621, 6.029474070169930, 7.267542695024334, 7.600272927483715, 7.951747596549428, 8.480014600428699, 9.308218360794886, 9.664585647411515, 10.39237496400697, 10.82841134168193, 11.66785735318537, 12.07108989685401, 12.28892480259116, 13.13063349813610, 13.84430612260735, 14.15008613204919, 14.56481279733384, 15.36285794550942
