| L(s) = 1 | − 8·4-s + 6·9-s + 40·16-s + 20·25-s − 48·36-s − 160·64-s + 27·81-s − 20·97-s − 160·100-s + 28·103-s + 127-s + 131-s + 137-s + 139-s + 240·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | − 4·4-s + 2·9-s + 10·16-s + 4·25-s − 8·36-s − 20·64-s + 3·81-s − 2.03·97-s − 16·100-s + 2.75·103-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 20·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9773238737\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9773238737\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 5 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 7 | $C_2^3$ | \( 1 + 23 T^{4} + p^{4} T^{8} \) |
| 13 | $C_2^3$ | \( 1 + 146 T^{4} + p^{4} T^{8} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 19 | $C_2^3$ | \( 1 - 601 T^{4} + p^{4} T^{8} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 + 59 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 47 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 43 | $C_2^3$ | \( 1 - 3214 T^{4} + p^{4} T^{8} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 61 | $C_2^3$ | \( 1 + 7199 T^{4} + p^{4} T^{8} \) |
| 67 | $C_2^2$ | \( ( 1 - 13 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 73 | $C_2^3$ | \( 1 + 9791 T^{4} + p^{4} T^{8} \) |
| 79 | $C_2^3$ | \( 1 - 12361 T^{4} + p^{4} T^{8} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.347956685847880986696871005034, −8.258438937499582565908932718424, −7.77508912595814080795782446503, −7.73329141881924292141229107787, −7.52335928053975352138779997158, −6.92972025779182455325444950926, −6.77998181941471786852283585538, −6.72862375338839652527160718751, −6.20014000464900308388985312388, −5.70241968590186156806073882705, −5.56036257637811136158288188543, −5.28156470869048448524062536264, −5.03082151355531916114211695144, −4.65006200484959930302109933552, −4.57415061272414722106561705141, −4.48712619187892255466753298233, −4.05960797364437868382973413759, −3.93407457127068675154333115218, −3.29643756510945456935248468194, −3.24603864210135940869137155000, −2.98746016356221503867303491852, −2.03748272588675044291118608869, −1.29696037014539780302674272649, −1.04584361655707915862064618320, −0.62291817646172884986176976294,
0.62291817646172884986176976294, 1.04584361655707915862064618320, 1.29696037014539780302674272649, 2.03748272588675044291118608869, 2.98746016356221503867303491852, 3.24603864210135940869137155000, 3.29643756510945456935248468194, 3.93407457127068675154333115218, 4.05960797364437868382973413759, 4.48712619187892255466753298233, 4.57415061272414722106561705141, 4.65006200484959930302109933552, 5.03082151355531916114211695144, 5.28156470869048448524062536264, 5.56036257637811136158288188543, 5.70241968590186156806073882705, 6.20014000464900308388985312388, 6.72862375338839652527160718751, 6.77998181941471786852283585538, 6.92972025779182455325444950926, 7.52335928053975352138779997158, 7.73329141881924292141229107787, 7.77508912595814080795782446503, 8.258438937499582565908932718424, 8.347956685847880986696871005034
