| L(s) = 1 | − 4-s − 9-s − 8·11-s + 16-s + 8·19-s + 4·29-s + 36-s − 12·41-s + 8·44-s − 49-s − 8·59-s + 12·61-s − 64-s + 16·71-s − 8·76-s + 81-s + 12·89-s + 8·99-s − 4·101-s + 4·109-s − 4·116-s + 26·121-s + 127-s + 131-s + 137-s + 139-s − 144-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1/3·9-s − 2.41·11-s + 1/4·16-s + 1.83·19-s + 0.742·29-s + 1/6·36-s − 1.87·41-s + 1.20·44-s − 1/7·49-s − 1.04·59-s + 1.53·61-s − 1/8·64-s + 1.89·71-s − 0.917·76-s + 1/9·81-s + 1.27·89-s + 0.804·99-s − 0.398·101-s + 0.383·109-s − 0.371·116-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.0833·144-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1102500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1102500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9947549106\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9947549106\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
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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
−10.08988142246171816333105146519, −9.878281140281778810296220485956, −9.307215598219349727443615852565, −8.858629048484985253080819050319, −8.381398093182565916602536198385, −8.078940799036683463191045508709, −7.55424875193128866131440014876, −7.54081455174277370117470627374, −6.69789449947503591538753446986, −6.38405499306493241796379451237, −5.51543246987784864238787750689, −5.43742236789017511637270918504, −4.88720046406530784640123264312, −4.80612288257266238837571564354, −3.68933291315861357170597846532, −3.46547048821906674206182966468, −2.69699799632198537109020192183, −2.48067351646546667041921647201, −1.43589860649440578340056872560, −0.46322738231351764285810569445,
0.46322738231351764285810569445, 1.43589860649440578340056872560, 2.48067351646546667041921647201, 2.69699799632198537109020192183, 3.46547048821906674206182966468, 3.68933291315861357170597846532, 4.80612288257266238837571564354, 4.88720046406530784640123264312, 5.43742236789017511637270918504, 5.51543246987784864238787750689, 6.38405499306493241796379451237, 6.69789449947503591538753446986, 7.54081455174277370117470627374, 7.55424875193128866131440014876, 8.078940799036683463191045508709, 8.381398093182565916602536198385, 8.858629048484985253080819050319, 9.307215598219349727443615852565, 9.878281140281778810296220485956, 10.08988142246171816333105146519
