| L(s) = 1 | + 2·3-s − 2·5-s + 3·9-s + 12·13-s − 4·15-s − 25-s + 4·27-s + 16·31-s + 4·37-s + 24·39-s − 4·41-s − 8·43-s − 6·45-s + 10·49-s − 12·53-s − 24·65-s − 16·67-s + 24·71-s − 2·75-s + 5·81-s + 8·83-s − 20·89-s + 32·93-s − 24·107-s + 8·111-s + 36·117-s + 18·121-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s + 9-s + 3.32·13-s − 1.03·15-s − 1/5·25-s + 0.769·27-s + 2.87·31-s + 0.657·37-s + 3.84·39-s − 0.624·41-s − 1.21·43-s − 0.894·45-s + 10/7·49-s − 1.64·53-s − 2.97·65-s − 1.95·67-s + 2.84·71-s − 0.230·75-s + 5/9·81-s + 0.878·83-s − 2.11·89-s + 3.31·93-s − 2.32·107-s + 0.759·111-s + 3.32·117-s + 1.63·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.891059824\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.891059824\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−8.513275540089964195522773002856, −8.443956919788316230091552925038, −8.108102182958424751085592907644, −7.75466662178013593451115977926, −7.33152834929353575735137745207, −6.83856423514350708286463306376, −6.40954027438300541165022039644, −6.11728987448086918048478547519, −6.02668805031417311521054762534, −5.13410895862985464579805844317, −4.83117280368541482849724808824, −4.25471601595203104119118644008, −3.88503491778614909349831885469, −3.74271592482130219761233705900, −3.27607771599875131833892687277, −2.84530438744157926399235612444, −2.41968875467660309684656623631, −1.42825744328556809689928239227, −1.40458780056428103856653163408, −0.64980809887545010900785055522,
0.64980809887545010900785055522, 1.40458780056428103856653163408, 1.42825744328556809689928239227, 2.41968875467660309684656623631, 2.84530438744157926399235612444, 3.27607771599875131833892687277, 3.74271592482130219761233705900, 3.88503491778614909349831885469, 4.25471601595203104119118644008, 4.83117280368541482849724808824, 5.13410895862985464579805844317, 6.02668805031417311521054762534, 6.11728987448086918048478547519, 6.40954027438300541165022039644, 6.83856423514350708286463306376, 7.33152834929353575735137745207, 7.75466662178013593451115977926, 8.108102182958424751085592907644, 8.443956919788316230091552925038, 8.513275540089964195522773002856
