| L(s) = 1 | + 2·5-s − 5·11-s − 4·23-s − 7·25-s − 14·31-s − 12·37-s − 12·47-s − 5·49-s − 10·53-s − 10·55-s + 8·59-s − 20·67-s + 16·71-s − 12·89-s − 2·97-s + 8·103-s + 12·113-s − 8·115-s + 14·121-s − 26·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 28·155-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1.50·11-s − 0.834·23-s − 7/5·25-s − 2.51·31-s − 1.97·37-s − 1.75·47-s − 5/7·49-s − 1.37·53-s − 1.34·55-s + 1.04·59-s − 2.44·67-s + 1.89·71-s − 1.27·89-s − 0.203·97-s + 0.788·103-s + 1.12·113-s − 0.746·115-s + 1.27·121-s − 2.32·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 2.24·155-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5645376 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5645376 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3168298615\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3168298615\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 11 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
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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
−7.28290021897970859593133073895, −6.92555186812727586069118504522, −6.37188903195207275650146805704, −5.83114431882706306108946061615, −5.79729187121046804129336776899, −5.18432945826177342955590271236, −5.02930381294477970138380086092, −4.44970129802461299250291539437, −3.73747142820058336171656741048, −3.48503979598989868638885321849, −2.96008069711960973958577277810, −2.27521607666270957351120023905, −1.73582051333148490489274132763, −1.72279183927496192016704996647, −0.17258259200106663880130073877,
0.17258259200106663880130073877, 1.72279183927496192016704996647, 1.73582051333148490489274132763, 2.27521607666270957351120023905, 2.96008069711960973958577277810, 3.48503979598989868638885321849, 3.73747142820058336171656741048, 4.44970129802461299250291539437, 5.02930381294477970138380086092, 5.18432945826177342955590271236, 5.79729187121046804129336776899, 5.83114431882706306108946061615, 6.37188903195207275650146805704, 6.92555186812727586069118504522, 7.28290021897970859593133073895
