| L(s) = 1 | − 4-s + 4·5-s − 4·11-s + 16-s − 4·20-s + 11·25-s − 16·31-s − 4·41-s + 4·44-s + 10·49-s − 16·55-s + 20·59-s + 4·61-s − 64-s − 24·71-s + 4·80-s − 20·89-s − 11·100-s + 16·101-s − 20·109-s − 10·121-s + 16·124-s + 24·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 1.78·5-s − 1.20·11-s + 1/4·16-s − 0.894·20-s + 11/5·25-s − 2.87·31-s − 0.624·41-s + 0.603·44-s + 10/7·49-s − 2.15·55-s + 2.60·59-s + 0.512·61-s − 1/8·64-s − 2.84·71-s + 0.447·80-s − 2.11·89-s − 1.09·100-s + 1.59·101-s − 1.91·109-s − 0.909·121-s + 1.43·124-s + 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.043426095\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.043426095\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 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 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−14.50199822172178467627442103768, −13.66947862458940057075093656648, −13.30763858333215636268292910519, −13.02095073289356055953262626066, −12.55301429059214250950799052292, −11.76599487031784910059380809884, −10.84254438299821749089591769568, −10.63040778928227546754185318849, −9.863274668603248122654645642351, −9.682713000609979304488212255995, −8.696974495503308049481832687830, −8.682179045626656078673976932616, −7.45510815631309209557330979476, −7.04343151172956597140121944009, −6.02605874746625091471776598664, −5.43274071978760288338344978568, −5.21328841124841593110153570342, −4.01139568069171076350526356250, −2.82749250776074870700688791646, −1.87973338753088038767096352128,
1.87973338753088038767096352128, 2.82749250776074870700688791646, 4.01139568069171076350526356250, 5.21328841124841593110153570342, 5.43274071978760288338344978568, 6.02605874746625091471776598664, 7.04343151172956597140121944009, 7.45510815631309209557330979476, 8.682179045626656078673976932616, 8.696974495503308049481832687830, 9.682713000609979304488212255995, 9.863274668603248122654645642351, 10.63040778928227546754185318849, 10.84254438299821749089591769568, 11.76599487031784910059380809884, 12.55301429059214250950799052292, 13.02095073289356055953262626066, 13.30763858333215636268292910519, 13.66947862458940057075093656648, 14.50199822172178467627442103768
