L(s) = 1 | + 4·7-s − 8·19-s + 10·25-s − 12·29-s − 8·31-s + 4·37-s + 9·49-s − 12·53-s − 24·59-s − 24·83-s − 8·103-s + 20·109-s + 12·113-s + 10·121-s + 127-s + 131-s − 32·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 26·169-s + 173-s + 40·175-s + ⋯ |
L(s) = 1 | + 1.51·7-s − 1.83·19-s + 2·25-s − 2.22·29-s − 1.43·31-s + 0.657·37-s + 9/7·49-s − 1.64·53-s − 3.12·59-s − 2.63·83-s − 0.788·103-s + 1.91·109-s + 1.12·113-s + 0.909·121-s + 0.0887·127-s + 0.0873·131-s − 2.77·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2·169-s + 0.0760·173-s + 3.02·175-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16257024 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16257024 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.593461707\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.593461707\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
good | 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
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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
−8.650160060374412327563999776077, −8.140236842863043939592997127714, −8.065586621456287072468669383376, −7.36692360576498340970342407118, −7.36596129621046186560102438549, −6.90942699995464603918491130614, −6.39055698309630441475480556291, −5.88882031785211984023459342865, −5.80215894323858054924559630081, −5.20186014165233210085862576695, −4.77859774358522471463383159836, −4.41821273692427832149287134388, −4.36361081922535137529644733429, −3.54179344349884505746062420163, −3.29200290169809319248342112313, −2.62137315460838839491748861683, −2.13203154132026791604788157824, −1.58150446048113745780406392725, −1.45914790337154626334973100742, −0.34813396758386217322458515654,
0.34813396758386217322458515654, 1.45914790337154626334973100742, 1.58150446048113745780406392725, 2.13203154132026791604788157824, 2.62137315460838839491748861683, 3.29200290169809319248342112313, 3.54179344349884505746062420163, 4.36361081922535137529644733429, 4.41821273692427832149287134388, 4.77859774358522471463383159836, 5.20186014165233210085862576695, 5.80215894323858054924559630081, 5.88882031785211984023459342865, 6.39055698309630441475480556291, 6.90942699995464603918491130614, 7.36596129621046186560102438549, 7.36692360576498340970342407118, 8.065586621456287072468669383376, 8.140236842863043939592997127714, 8.650160060374412327563999776077