| L(s) = 1 | + 2-s + 3-s + 4-s + 3·5-s + 6-s + 8-s + 9-s + 3·10-s + 12-s + 3·15-s + 16-s + 18-s − 2·19-s + 3·20-s − 6·23-s + 24-s + 8·25-s + 27-s + 5·29-s + 3·30-s + 32-s + 36-s − 2·38-s + 3·40-s + 8·43-s + 3·45-s − 6·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.34·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.948·10-s + 0.288·12-s + 0.774·15-s + 1/4·16-s + 0.235·18-s − 0.458·19-s + 0.670·20-s − 1.25·23-s + 0.204·24-s + 8/5·25-s + 0.192·27-s + 0.928·29-s + 0.547·30-s + 0.176·32-s + 1/6·36-s − 0.324·38-s + 0.474·40-s + 1.21·43-s + 0.447·45-s − 0.884·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 501120 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 501120 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.135691859\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.135691859\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 3 | $C_1$ | \( 1 - T \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 4 T + p T^{2} ) \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 6 T + p T^{2} ) \) |
| good | 7 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 44 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 10 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 76 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 122 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−8.539143577316954500917777602858, −8.160877622279476568476731904211, −7.41028450772846461902078714925, −7.18391827876252656053093999714, −6.54851657101329529787635822845, −6.10333025604841291853794774055, −5.82396713473896062460771430104, −5.25104049617269816213368727885, −4.73071220477823842352457417954, −4.13135340034110495561095848037, −3.74448040809911503075336081541, −2.82404626401425922567103813509, −2.50080725721222916370773799781, −1.94062746768802502427067171010, −1.11506285716877831610700158396,
1.11506285716877831610700158396, 1.94062746768802502427067171010, 2.50080725721222916370773799781, 2.82404626401425922567103813509, 3.74448040809911503075336081541, 4.13135340034110495561095848037, 4.73071220477823842352457417954, 5.25104049617269816213368727885, 5.82396713473896062460771430104, 6.10333025604841291853794774055, 6.54851657101329529787635822845, 7.18391827876252656053093999714, 7.41028450772846461902078714925, 8.160877622279476568476731904211, 8.539143577316954500917777602858
