| L(s) = 1 | − 9-s − 12·17-s − 25-s − 16·31-s + 12·41-s + 16·47-s − 14·49-s − 16·71-s + 12·73-s − 16·79-s + 81-s − 20·89-s + 4·97-s − 12·113-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 12·153-s + 157-s + 163-s + 167-s − 10·169-s + 173-s + ⋯ |
| L(s) = 1 | − 1/3·9-s − 2.91·17-s − 1/5·25-s − 2.87·31-s + 1.87·41-s + 2.33·47-s − 2·49-s − 1.89·71-s + 1.40·73-s − 1.80·79-s + 1/9·81-s − 2.11·89-s + 0.406·97-s − 1.12·113-s + 6/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.970·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.769·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 2 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−8.500325317326143307596410904104, −7.912796599625954559781492977497, −7.45780800984041549694161985168, −7.18617849035328307979332807699, −6.97877706390390097886219572548, −6.42206554088495430335224426829, −5.93975246916964310309873161845, −5.93266186724602601673909692579, −5.25105498951903355415298803549, −4.94089114047611548370176630028, −4.41252551121001951351896294295, −4.02029321570309562791742492301, −3.88078615245765628348069932634, −3.17169513287499185040718865944, −2.48764007961521057354294396647, −2.42346376613088842617251768509, −1.79158125388477419855318330873, −1.22354189487460569191438815017, 0, 0,
1.22354189487460569191438815017, 1.79158125388477419855318330873, 2.42346376613088842617251768509, 2.48764007961521057354294396647, 3.17169513287499185040718865944, 3.88078615245765628348069932634, 4.02029321570309562791742492301, 4.41252551121001951351896294295, 4.94089114047611548370176630028, 5.25105498951903355415298803549, 5.93266186724602601673909692579, 5.93975246916964310309873161845, 6.42206554088495430335224426829, 6.97877706390390097886219572548, 7.18617849035328307979332807699, 7.45780800984041549694161985168, 7.912796599625954559781492977497, 8.500325317326143307596410904104
