| L(s) = 1 | − 2·5-s − 2·9-s + 3·25-s − 4·37-s + 4·45-s − 2·49-s − 4·53-s + 3·81-s − 4·89-s − 121-s − 4·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 8·185-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | − 2·5-s − 2·9-s + 3·25-s − 4·37-s + 4·45-s − 2·49-s − 4·53-s + 3·81-s − 4·89-s − 121-s − 4·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 8·185-s + 191-s + 193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12390400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12390400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.09049949392\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09049949392\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 37 | $C_1$ | \( ( 1 + T )^{4} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 53 | $C_1$ | \( ( 1 + T )^{4} \) |
| 59 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_1$ | \( ( 1 + T )^{4} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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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
−8.906272173887560652249424208287, −8.391125004428543911437387827701, −8.239777849152239474998307888799, −7.904789609333678077107079491194, −7.70361026860171457198786144363, −6.89215499809786808363089351634, −6.88052814560047899578070815707, −6.51000948881895579438659990939, −5.93019594615811831594958349597, −5.36136573580389731364830629085, −5.22799277169442149717506573343, −4.71755390172079802577866346334, −4.38005243966857215626882863804, −3.76688908177301172060046679210, −3.33632972757570360585191736771, −3.04826805153953325300783283596, −2.96693692365797606635619518738, −1.94035255978083368585762446169, −1.42428804254149585396632902344, −0.17542455797828592980790210424,
0.17542455797828592980790210424, 1.42428804254149585396632902344, 1.94035255978083368585762446169, 2.96693692365797606635619518738, 3.04826805153953325300783283596, 3.33632972757570360585191736771, 3.76688908177301172060046679210, 4.38005243966857215626882863804, 4.71755390172079802577866346334, 5.22799277169442149717506573343, 5.36136573580389731364830629085, 5.93019594615811831594958349597, 6.51000948881895579438659990939, 6.88052814560047899578070815707, 6.89215499809786808363089351634, 7.70361026860171457198786144363, 7.904789609333678077107079491194, 8.239777849152239474998307888799, 8.391125004428543911437387827701, 8.906272173887560652249424208287
