| L(s) = 1 | + 4·4-s + 9·5-s + 4·16-s + 36·20-s + 40·25-s + 5·31-s + 14·37-s − 36·47-s − 14·49-s − 45·59-s − 16·64-s − 13·67-s + 36·80-s + 17·97-s + 160·100-s − 4·103-s + 63·113-s + 11·121-s + 20·124-s + 117·125-s + 127-s + 131-s + 137-s + 139-s + 56·148-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 2·4-s + 4.02·5-s + 16-s + 8.04·20-s + 8·25-s + 0.898·31-s + 2.30·37-s − 5.25·47-s − 2·49-s − 5.85·59-s − 2·64-s − 1.58·67-s + 4.02·80-s + 1.72·97-s + 16·100-s − 0.394·103-s + 5.92·113-s + 121-s + 1.79·124-s + 10.4·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 4.60·148-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.163818533\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.163818533\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| good | 2 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 5 | $C_2$$\times$$C_2^2$ | \( ( 1 - 3 T + p T^{2} )^{2}( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} ) \) |
| 7 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 23 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 9 T + 58 T^{2} - 9 p T^{3} + p^{2} T^{4} )( 1 + 9 T + 58 T^{2} + 9 p T^{3} + p^{2} T^{4} ) \) |
| 29 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$$\times$$C_2^2$ | \( ( 1 - 5 T + p T^{2} )^{2}( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} ) \) |
| 37 | $C_2^2$ | \( ( 1 - 7 T + 12 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2$$\times$$C_2^2$ | \( ( 1 + 12 T + p T^{2} )^{2}( 1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4} ) \) |
| 53 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \) |
| 59 | $C_2$$\times$$C_2^2$ | \( ( 1 + 15 T + p T^{2} )^{2}( 1 + 15 T + 166 T^{2} + 15 p T^{3} + p^{2} T^{4} ) \) |
| 61 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 + 13 T + p T^{2} )^{2}( 1 - 13 T + 102 T^{2} - 13 p T^{3} + p^{2} T^{4} ) \) |
| 71 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 3 T - 62 T^{2} - 3 p T^{3} + p^{2} T^{4} )( 1 + 3 T - 62 T^{2} + 3 p T^{3} + p^{2} T^{4} ) \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2}( 1 + 9 T + p T^{2} )^{2} \) |
| 97 | $C_2$$\times$$C_2^2$ | \( ( 1 - 17 T + p T^{2} )^{2}( 1 + 17 T + 192 T^{2} + 17 p T^{3} + p^{2} T^{4} ) \) |
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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
−8.796452325566257997652107630784, −8.166278873556751261167502675153, −7.963237173800474893054123151582, −7.69372026952203388279995885158, −7.68603430044049099713195368604, −7.02184544327875127960369739447, −6.77295871758263049816655760014, −6.54408859185573138809475425726, −6.49932726519668840934007904059, −6.07981394777200980670435904824, −5.97397549324412686872515571821, −5.92553546542967692245899095663, −5.78052714732256427149376560786, −4.96337395849272389746786223983, −4.80512946652009250519402585032, −4.60853041874035947972088376335, −4.58869607639560251570757244828, −3.39780115576335479465276021166, −3.04945806214358337515940235456, −2.95127816312683059618374824394, −2.75518159811523909875378913338, −1.92744279389285753421860503538, −1.88563412391320667062298837421, −1.79500035656915775258886071133, −1.40501905323896177182896856443,
1.40501905323896177182896856443, 1.79500035656915775258886071133, 1.88563412391320667062298837421, 1.92744279389285753421860503538, 2.75518159811523909875378913338, 2.95127816312683059618374824394, 3.04945806214358337515940235456, 3.39780115576335479465276021166, 4.58869607639560251570757244828, 4.60853041874035947972088376335, 4.80512946652009250519402585032, 4.96337395849272389746786223983, 5.78052714732256427149376560786, 5.92553546542967692245899095663, 5.97397549324412686872515571821, 6.07981394777200980670435904824, 6.49932726519668840934007904059, 6.54408859185573138809475425726, 6.77295871758263049816655760014, 7.02184544327875127960369739447, 7.68603430044049099713195368604, 7.69372026952203388279995885158, 7.963237173800474893054123151582, 8.166278873556751261167502675153, 8.796452325566257997652107630784
