| L(s) = 1 | + 4·5-s + 6·11-s − 4·16-s + 11·25-s + 2·29-s − 16·31-s − 14·41-s + 10·49-s + 24·55-s + 20·59-s + 4·61-s + 16·71-s + 20·79-s − 16·80-s + 20·89-s − 34·101-s + 10·109-s + 5·121-s + 24·125-s + 127-s + 131-s + 137-s + 139-s + 8·145-s + 149-s + 151-s − 64·155-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 1.80·11-s − 16-s + 11/5·25-s + 0.371·29-s − 2.87·31-s − 2.18·41-s + 10/7·49-s + 3.23·55-s + 2.60·59-s + 0.512·61-s + 1.89·71-s + 2.25·79-s − 1.78·80-s + 2.11·89-s − 3.38·101-s + 0.957·109-s + 5/11·121-s + 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.664·145-s + 0.0819·149-s + 0.0813·151-s − 5.14·155-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1703025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1703025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.593821200\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.593821200\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 29 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 145 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
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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
−9.757830890324680421257500645776, −9.304651197910903090142827882416, −9.087984649008141095164856247320, −8.943330686626200754391810227476, −8.431790346998482419687020396985, −7.83505957045947584722038934808, −7.16829402594956136890185706198, −6.80125036026447721708620469006, −6.52940572166120069586844787306, −6.36868223560415642204125320538, −5.50795266801216663163731395857, −5.27356761140796604447676436149, −5.09538745232020226661396787702, −4.02770147782157486020210128698, −3.93877574367662835515018825413, −3.31025327241847479150162293105, −2.46081331880117024809509595742, −1.98979331973206540154965494133, −1.66837869397666754533906565074, −0.808983604750814900464099448857,
0.808983604750814900464099448857, 1.66837869397666754533906565074, 1.98979331973206540154965494133, 2.46081331880117024809509595742, 3.31025327241847479150162293105, 3.93877574367662835515018825413, 4.02770147782157486020210128698, 5.09538745232020226661396787702, 5.27356761140796604447676436149, 5.50795266801216663163731395857, 6.36868223560415642204125320538, 6.52940572166120069586844787306, 6.80125036026447721708620469006, 7.16829402594956136890185706198, 7.83505957045947584722038934808, 8.431790346998482419687020396985, 8.943330686626200754391810227476, 9.087984649008141095164856247320, 9.304651197910903090142827882416, 9.757830890324680421257500645776
