| L(s) = 1 | + 4·4-s + 9-s − 6·11-s + 12·16-s − 5·25-s + 18·29-s + 4·36-s − 24·44-s + 32·64-s − 24·71-s + 2·79-s − 8·81-s − 6·99-s − 20·100-s + 22·109-s + 72·116-s + 5·121-s + 127-s + 131-s + 137-s + 139-s + 12·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 2·4-s + 1/3·9-s − 1.80·11-s + 3·16-s − 25-s + 3.34·29-s + 2/3·36-s − 3.61·44-s + 4·64-s − 2.84·71-s + 0.225·79-s − 8/9·81-s − 0.603·99-s − 2·100-s + 2.10·109-s + 6.68·116-s + 5/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.180394812\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.180394812\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_2$ | \( 1 + p T^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 3 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 29 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 31 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 149 T^{2} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.10567468930279138081106970108, −12.00811015690958817629287232804, −11.30235918744439738653927712257, −10.94043815523613723900054249655, −10.37362848666299678503040484261, −10.09055563370478056023082264713, −9.930643674479062817701754285512, −8.683589532414420174249203549147, −8.328075260023874430641770387966, −7.72085147515495833260072374689, −7.41486809145394920894333838278, −6.91681542274611530157447741136, −6.15428660163534749651699284419, −6.02230600328434106335755827916, −5.16345015833196449681522128795, −4.59936133963148349976905846613, −3.50979834071970860028885937728, −2.69472791364458900492463139387, −2.51035181150201167130371088023, −1.37714161154984888326967714381,
1.37714161154984888326967714381, 2.51035181150201167130371088023, 2.69472791364458900492463139387, 3.50979834071970860028885937728, 4.59936133963148349976905846613, 5.16345015833196449681522128795, 6.02230600328434106335755827916, 6.15428660163534749651699284419, 6.91681542274611530157447741136, 7.41486809145394920894333838278, 7.72085147515495833260072374689, 8.328075260023874430641770387966, 8.683589532414420174249203549147, 9.930643674479062817701754285512, 10.09055563370478056023082264713, 10.37362848666299678503040484261, 10.94043815523613723900054249655, 11.30235918744439738653927712257, 12.00811015690958817629287232804, 12.10567468930279138081106970108
