| L(s) = 1 | − 2·2-s − 4-s + 3·5-s + 8·8-s − 6·10-s + 3·11-s − 3·13-s − 7·16-s − 2·19-s − 3·20-s − 6·22-s − 2·23-s + 25-s + 6·26-s + 2·29-s − 9·31-s − 14·32-s + 9·37-s + 4·38-s + 24·40-s − 11·41-s + 8·43-s − 3·44-s + 4·46-s − 6·47-s − 14·49-s − 2·50-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1/2·4-s + 1.34·5-s + 2.82·8-s − 1.89·10-s + 0.904·11-s − 0.832·13-s − 7/4·16-s − 0.458·19-s − 0.670·20-s − 1.27·22-s − 0.417·23-s + 1/5·25-s + 1.17·26-s + 0.371·29-s − 1.61·31-s − 2.47·32-s + 1.47·37-s + 0.648·38-s + 3.79·40-s − 1.71·41-s + 1.21·43-s − 0.452·44-s + 0.589·46-s − 0.875·47-s − 2·49-s − 0.282·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36036009 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36036009 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 + T )^{2} \) |
| 29 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - 3 T + 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $D_{4}$ | \( 1 - 3 T + 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 3 T + 24 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 9 T + 78 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 9 T + 90 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 11 T + 108 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 8 T + 34 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 6 T + 86 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 15 T + 170 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 3 T + 18 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 17 T + 202 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 25 T + 294 T^{2} + 25 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 2 T + 130 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 - 6 T + 158 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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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
−7.910580099573488211738702404049, −7.69672575415617874989788090750, −7.45092790700056853720250896959, −6.92013375058656279810116379145, −6.43397134591433395658737814711, −6.33755197153873944713935072660, −5.62316983099132386787689076378, −5.57181464983287422614651152201, −4.84517885518254766041930415841, −4.79478620579140083835795130352, −4.23932622004694222893386678951, −4.05480843483099138363700848678, −3.30507888668318548641400306258, −3.05030636398238243356873778693, −2.11312584945081072998564455562, −1.91800926960019710086621260958, −1.47155029626420119034972834326, −1.10480956395666603084200917221, 0, 0,
1.10480956395666603084200917221, 1.47155029626420119034972834326, 1.91800926960019710086621260958, 2.11312584945081072998564455562, 3.05030636398238243356873778693, 3.30507888668318548641400306258, 4.05480843483099138363700848678, 4.23932622004694222893386678951, 4.79478620579140083835795130352, 4.84517885518254766041930415841, 5.57181464983287422614651152201, 5.62316983099132386787689076378, 6.33755197153873944713935072660, 6.43397134591433395658737814711, 6.92013375058656279810116379145, 7.45092790700056853720250896959, 7.69672575415617874989788090750, 7.910580099573488211738702404049
