| L(s) = 1 | + 4-s + 5·9-s + 6·11-s + 16-s − 2·25-s + 6·29-s + 5·36-s − 2·37-s − 12·43-s + 6·44-s + 64-s − 4·67-s + 6·71-s + 9·79-s + 16·81-s + 30·99-s − 2·100-s + 6·107-s − 6·109-s + 9·113-s + 6·116-s + 25·121-s + 127-s + 131-s + 137-s + 139-s + 5·144-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 5/3·9-s + 1.80·11-s + 1/4·16-s − 2/5·25-s + 1.11·29-s + 5/6·36-s − 0.328·37-s − 1.82·43-s + 0.904·44-s + 1/8·64-s − 0.488·67-s + 0.712·71-s + 1.01·79-s + 16/9·81-s + 3.01·99-s − 1/5·100-s + 0.580·107-s − 0.574·109-s + 0.846·113-s + 0.557·116-s + 2.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 5/12·144-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1162084 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1162084 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.472225579\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.472225579\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | | \( 1 \) |
| 11 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 23 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 35 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 35 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 35 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 71 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 143 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
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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
−8.081078843453114932365489248854, −7.40964628891965176435188151100, −7.17085081288222948362782559516, −6.63676457790334775624131151776, −6.48186608658561181970871658571, −6.08075294880438178112826746569, −5.30418516764757583867614470852, −4.79603712547085432561880203203, −4.40744018923658496387917148151, −3.79095818902129493363256903544, −3.59149871251627751883157105082, −2.82215452680288883413181490015, −1.93496934573361604221514240850, −1.56948866689806376200060301045, −0.920181782416923008352170535454,
0.920181782416923008352170535454, 1.56948866689806376200060301045, 1.93496934573361604221514240850, 2.82215452680288883413181490015, 3.59149871251627751883157105082, 3.79095818902129493363256903544, 4.40744018923658496387917148151, 4.79603712547085432561880203203, 5.30418516764757583867614470852, 6.08075294880438178112826746569, 6.48186608658561181970871658571, 6.63676457790334775624131151776, 7.17085081288222948362782559516, 7.40964628891965176435188151100, 8.081078843453114932365489248854
