| L(s) = 1 | + 10·7-s − 3·9-s + 4·16-s + 14·19-s − 14·31-s + 2·37-s + 6·43-s + 59·49-s − 30·63-s − 32·67-s + 34·73-s + 38·97-s + 38·109-s + 40·112-s + 127-s + 131-s + 140·133-s + 137-s + 139-s − 12·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s − 42·171-s + ⋯ |
| L(s) = 1 | + 3.77·7-s − 9-s + 16-s + 3.21·19-s − 2.51·31-s + 0.328·37-s + 0.914·43-s + 59/7·49-s − 3.77·63-s − 3.90·67-s + 3.97·73-s + 3.85·97-s + 3.63·109-s + 3.77·112-s + 0.0887·127-s + 0.0873·131-s + 12.1·133-s + 0.0854·137-s + 0.0848·139-s − 144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1/13·169-s − 3.21·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.611072417\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.611072417\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 5 | | \( 1 \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| good | 2 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} )( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} ) \) |
| 7 | $C_2$$\times$$C_2^2$ | \( ( 1 - 5 T + p T^{2} )^{2}( 1 + 2 T^{2} + p^{2} T^{4} ) \) |
| 11 | $C_2^3$ | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$$\times$$C_2^2$ | \( ( 1 - 7 T + p T^{2} )^{2}( 1 - 37 T^{2} + p^{2} T^{4} ) \) |
| 23 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$$\times$$C_2^2$ | \( ( 1 + 7 T + p T^{2} )^{2}( 1 - 13 T^{2} + p^{2} T^{4} ) \) |
| 37 | $C_2$$\times$$C_2^2$ | \( ( 1 - T + p T^{2} )^{2}( 1 + 47 T^{2} + p^{2} T^{4} ) \) |
| 41 | $C_2^3$ | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 5 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 59 | $C_2^3$ | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| 61 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 121 T^{2} + p^{2} T^{4} )( 1 + 74 T^{2} + p^{2} T^{4} ) \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 + 16 T + p T^{2} )^{2}( 1 - 109 T^{2} + p^{2} T^{4} ) \) |
| 71 | $C_2^3$ | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| 73 | $C_2$$\times$$C_2^2$ | \( ( 1 - 17 T + p T^{2} )^{2}( 1 + 143 T^{2} + p^{2} T^{4} ) \) |
| 79 | $C_2^2$ | \( ( 1 + 11 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| 97 | $C_2$$\times$$C_2^2$ | \( ( 1 - 19 T + p T^{2} )^{2}( 1 + 2 T^{2} + p^{2} T^{4} ) \) |
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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
−7.40913783440112748837378473886, −7.14552604719156323297839360948, −6.87088306750295836981745722993, −6.46435985444908508362497468530, −6.12080728681026397049640350647, −5.72957626854604558177339413545, −5.64294962458155539760536222082, −5.56135263228598417491645642356, −5.54516266061639270922907012687, −4.95983855346961438010963559312, −4.85369393511604900733344183717, −4.73625580593607963314510702454, −4.64709422296980249395175205425, −4.06620227419204619778280096180, −3.87507844688402859757388380279, −3.42163241250758204972069436897, −3.41717399633498239159076489432, −3.05433005084400286477855878564, −2.71659025428272463979998148386, −2.11345412132740137357206902414, −2.01256978401867072375286369376, −1.84571881612551019092567895965, −1.26723171417634862529441970980, −1.04599128683948065967689421348, −0.72247139631301541794106574053,
0.72247139631301541794106574053, 1.04599128683948065967689421348, 1.26723171417634862529441970980, 1.84571881612551019092567895965, 2.01256978401867072375286369376, 2.11345412132740137357206902414, 2.71659025428272463979998148386, 3.05433005084400286477855878564, 3.41717399633498239159076489432, 3.42163241250758204972069436897, 3.87507844688402859757388380279, 4.06620227419204619778280096180, 4.64709422296980249395175205425, 4.73625580593607963314510702454, 4.85369393511604900733344183717, 4.95983855346961438010963559312, 5.54516266061639270922907012687, 5.56135263228598417491645642356, 5.64294962458155539760536222082, 5.72957626854604558177339413545, 6.12080728681026397049640350647, 6.46435985444908508362497468530, 6.87088306750295836981745722993, 7.14552604719156323297839360948, 7.40913783440112748837378473886
