| L(s) = 1 | − 4·4-s + 10·9-s − 40·13-s + 12·16-s − 64·19-s − 128·31-s − 40·36-s − 80·37-s − 80·43-s + 14·49-s + 160·52-s − 56·61-s − 32·64-s − 240·67-s − 120·73-s + 256·76-s − 128·79-s + 19·81-s + 360·97-s + 160·103-s − 72·109-s − 400·117-s + 272·121-s + 512·124-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 4-s + 10/9·9-s − 3.07·13-s + 3/4·16-s − 3.36·19-s − 4.12·31-s − 1.11·36-s − 2.16·37-s − 1.86·43-s + 2/7·49-s + 3.07·52-s − 0.918·61-s − 1/2·64-s − 3.58·67-s − 1.64·73-s + 3.36·76-s − 1.62·79-s + 0.234·81-s + 3.71·97-s + 1.55·103-s − 0.660·109-s − 3.41·117-s + 2.24·121-s + 4.12·124-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.01305824424\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01305824424\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 3 | $C_2^2$ | \( 1 - 10 T^{2} + p^{4} T^{4} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| good | 11 | $D_4\times C_2$ | \( 1 - 272 T^{2} + 36578 T^{4} - 272 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 20 T + 326 T^{2} + 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 440 T^{2} + 204242 T^{4} - 440 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $C_2$ | \( ( 1 + 16 T + p^{2} T^{2} )^{4} \) |
| 23 | $D_4\times C_2$ | \( 1 + 688 T^{2} + 666818 T^{4} + 688 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 1652 T^{2} + 1917638 T^{4} - 1652 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 64 T + 2246 T^{2} + 64 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 20 T + p^{2} T^{2} )^{4} \) |
| 41 | $D_4\times C_2$ | \( 1 - 856 T^{2} - 2330094 T^{4} - 856 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 40 T + 3090 T^{2} + 40 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 4346 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 3026 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 10532 T^{2} + 49098278 T^{4} - 10532 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 28 T + 4838 T^{2} + 28 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 + 120 T + 12466 T^{2} + 120 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 12176 T^{2} + 74167106 T^{4} - 12176 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 60 T + 9766 T^{2} + 60 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 64 T + 10706 T^{2} + 64 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 6356 T^{2} - 6983674 T^{4} - 6356 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 27832 T^{2} + 318232338 T^{4} - 27832 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 180 T + 26470 T^{2} - 180 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−7.02309269626274293473007712241, −6.97610517770284146105974210929, −6.26813292943912495949421684718, −6.19634781315296470795389560608, −6.09077567028726848887316995808, −5.63726830822751839991483583358, −5.58956654883324141605930513718, −5.23422458914180974777171578042, −4.89285594895737962145209850532, −4.61068612317137178935944234056, −4.60360194904582596936579241115, −4.54501983204119656618655362791, −4.32252579339097288972488757454, −3.65928936748217591796796224161, −3.62334800887389138336154612765, −3.44753062455993237047697742474, −3.14487399302165315813810937895, −2.63672426163067867782711062485, −2.22622858282203119784563700600, −2.07686018973427293456721163400, −1.82276463167230130071608666968, −1.67046597437368885235711929010, −1.14319323700120410157533723690, −0.12487072401314354568837489369, −0.089323955795325810027217456560,
0.089323955795325810027217456560, 0.12487072401314354568837489369, 1.14319323700120410157533723690, 1.67046597437368885235711929010, 1.82276463167230130071608666968, 2.07686018973427293456721163400, 2.22622858282203119784563700600, 2.63672426163067867782711062485, 3.14487399302165315813810937895, 3.44753062455993237047697742474, 3.62334800887389138336154612765, 3.65928936748217591796796224161, 4.32252579339097288972488757454, 4.54501983204119656618655362791, 4.60360194904582596936579241115, 4.61068612317137178935944234056, 4.89285594895737962145209850532, 5.23422458914180974777171578042, 5.58956654883324141605930513718, 5.63726830822751839991483583358, 6.09077567028726848887316995808, 6.19634781315296470795389560608, 6.26813292943912495949421684718, 6.97610517770284146105974210929, 7.02309269626274293473007712241
