| L(s) = 1 | + 2·2-s + 3·4-s + 4·8-s + 5·16-s − 4·19-s + 2·25-s + 6·32-s − 8·38-s + 2·49-s + 4·50-s + 7·64-s − 12·76-s − 81-s + 4·98-s + 6·100-s + 127-s + 8·128-s + 131-s + 137-s + 139-s + 149-s + 151-s − 16·152-s + 157-s − 2·162-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 2·2-s + 3·4-s + 4·8-s + 5·16-s − 4·19-s + 2·25-s + 6·32-s − 8·38-s + 2·49-s + 4·50-s + 7·64-s − 12·76-s − 81-s + 4·98-s + 6·100-s + 127-s + 8·128-s + 131-s + 137-s + 139-s + 149-s + 151-s − 16·152-s + 157-s − 2·162-s + 163-s + 167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5345344 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5345344 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(5.014745264\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.014745264\) |
| \(L(1)\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 17 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + T^{4} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 11 | $C_2^2$ | \( 1 + T^{4} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 19 | $C_1$ | \( ( 1 + T )^{4} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_2^2$ | \( 1 + T^{4} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 67 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_2^2$ | \( 1 + T^{4} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + T^{4} \) |
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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
−9.262721823234324910135023539174, −8.883398798019097365126161524759, −8.385706739000222452975211300862, −8.337764998389182441896746949134, −7.66365608714956879840364692553, −7.13817165575138674566917653131, −6.93718979181982970125104715823, −6.51010807287264474384057637097, −6.13429333521976490677472215548, −6.01870069188019296488956209307, −5.26022981619227638648194666518, −5.02421312217761688986279394215, −4.41273145556537044693568969653, −4.27336608149448908364295689385, −3.88872460093310303424831648782, −3.33782721863083347108895231418, −2.60448804570589962107593101612, −2.47035804445673483457675783709, −1.94518733367350767376102188026, −1.20445136071991780073158345599,
1.20445136071991780073158345599, 1.94518733367350767376102188026, 2.47035804445673483457675783709, 2.60448804570589962107593101612, 3.33782721863083347108895231418, 3.88872460093310303424831648782, 4.27336608149448908364295689385, 4.41273145556537044693568969653, 5.02421312217761688986279394215, 5.26022981619227638648194666518, 6.01870069188019296488956209307, 6.13429333521976490677472215548, 6.51010807287264474384057637097, 6.93718979181982970125104715823, 7.13817165575138674566917653131, 7.66365608714956879840364692553, 8.337764998389182441896746949134, 8.385706739000222452975211300862, 8.883398798019097365126161524759, 9.262721823234324910135023539174
