| L(s) = 1 | − 3-s − 3·5-s + 3·9-s − 8·11-s + 5·13-s + 3·15-s + 5·17-s + 8·19-s + 23-s + 5·25-s − 8·27-s − 3·29-s + 8·31-s + 8·33-s + 4·37-s − 5·39-s + 5·41-s + 11·43-s − 9·45-s + 5·47-s − 14·49-s − 5·51-s + 9·53-s + 24·55-s − 8·57-s − 13·59-s + 61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34·5-s + 9-s − 2.41·11-s + 1.38·13-s + 0.774·15-s + 1.21·17-s + 1.83·19-s + 0.208·23-s + 25-s − 1.53·27-s − 0.557·29-s + 1.43·31-s + 1.39·33-s + 0.657·37-s − 0.800·39-s + 0.780·41-s + 1.67·43-s − 1.34·45-s + 0.729·47-s − 2·49-s − 0.700·51-s + 1.23·53-s + 3.23·55-s − 1.05·57-s − 1.69·59-s + 0.128·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23104 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23104 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7801584269\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7801584269\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( 1 - 8 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 5 T + 8 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - T - 22 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 5 T - 16 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 11 T + 78 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 5 T - 22 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 9 T + 28 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 13 T + 110 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + T - 70 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 9 T + 8 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 3 T - 80 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 13 T + 72 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
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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
−13.49623060698242654792520311593, −12.78901816022090662434005503767, −12.01255057718805074530326390348, −11.92785765795071117290117393839, −11.13253141962830958834015744702, −10.80376756342128163847906372351, −10.46177320669096020045510745729, −9.656916818908247081094457171789, −9.367134985478709320882469849148, −8.216548688067344586752233555432, −7.78111380016531325454854692236, −7.75348182049944111759256174068, −7.09112036315099228024375911223, −6.09566197658560200325766043446, −5.53051998006997938078980694943, −5.03494778594923827488258126283, −4.22027592841996732724125269361, −3.50875417154481424874690094537, −2.78405572274053258556836322948, −0.979914450721557854737958164682,
0.979914450721557854737958164682, 2.78405572274053258556836322948, 3.50875417154481424874690094537, 4.22027592841996732724125269361, 5.03494778594923827488258126283, 5.53051998006997938078980694943, 6.09566197658560200325766043446, 7.09112036315099228024375911223, 7.75348182049944111759256174068, 7.78111380016531325454854692236, 8.216548688067344586752233555432, 9.367134985478709320882469849148, 9.656916818908247081094457171789, 10.46177320669096020045510745729, 10.80376756342128163847906372351, 11.13253141962830958834015744702, 11.92785765795071117290117393839, 12.01255057718805074530326390348, 12.78901816022090662434005503767, 13.49623060698242654792520311593
