| L(s) = 1 | + 9·9-s − 6·19-s + 14·29-s + 12·31-s − 44·41-s + 9·49-s − 22·59-s − 32·61-s − 52·79-s + 34·81-s + 12·89-s − 8·101-s − 6·109-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 11·169-s − 54·171-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | + 3·9-s − 1.37·19-s + 2.59·29-s + 2.15·31-s − 6.87·41-s + 9/7·49-s − 2.86·59-s − 4.09·61-s − 5.85·79-s + 34/9·81-s + 1.27·89-s − 0.796·101-s − 0.574·109-s − 0.909·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.846·169-s − 4.12·171-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.847515695\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.847515695\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( ( 1 + T )^{6} \) |
| good | 3 | \( 1 - p^{2} T^{2} + 47 T^{4} - 170 T^{6} + 47 p^{2} T^{8} - p^{6} T^{10} + p^{6} T^{12} \) |
| 7 | \( 1 - 9 T^{2} + 5 p T^{4} - 38 T^{6} + 5 p^{3} T^{8} - 9 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( ( 1 + 5 T^{2} + 16 T^{3} + 5 p T^{4} + p^{3} T^{6} )^{2} \) |
| 13 | \( 1 + 11 T^{2} + 55 T^{4} - 2146 T^{6} + 55 p^{2} T^{8} + 11 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( 1 - 5 p T^{2} + 3219 T^{4} - 70126 T^{6} + 3219 p^{2} T^{8} - 5 p^{5} T^{10} + p^{6} T^{12} \) |
| 23 | \( 1 - 97 T^{2} + 4419 T^{4} - 124918 T^{6} + 4419 p^{2} T^{8} - 97 p^{4} T^{10} + p^{6} T^{12} \) |
| 29 | \( ( 1 - 7 T + 43 T^{2} - 114 T^{3} + 43 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( ( 1 - 6 T + 77 T^{2} - 340 T^{3} + 77 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( 1 - 2 p T^{2} + 4747 T^{4} - 190436 T^{6} + 4747 p^{2} T^{8} - 2 p^{5} T^{10} + p^{6} T^{12} \) |
| 41 | \( ( 1 + 22 T + 223 T^{2} + 1572 T^{3} + 223 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( 1 - 94 T^{2} + 7591 T^{4} - 340324 T^{6} + 7591 p^{2} T^{8} - 94 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( 1 - 58 T^{2} + 3567 T^{4} - 270316 T^{6} + 3567 p^{2} T^{8} - 58 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( 1 + 67 T^{2} + 7527 T^{4} + 328942 T^{6} + 7527 p^{2} T^{8} + 67 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( ( 1 + 11 T + 37 T^{2} - 246 T^{3} + 37 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 61 | \( ( 1 + 16 T + 207 T^{2} + 1600 T^{3} + 207 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 - 3 p T^{2} + 22943 T^{4} - 1802666 T^{6} + 22943 p^{2} T^{8} - 3 p^{5} T^{10} + p^{6} T^{12} \) |
| 71 | \( ( 1 + p T^{2} )^{6} \) |
| 73 | \( 1 - 69 T^{2} + 14051 T^{4} - 586094 T^{6} + 14051 p^{2} T^{8} - 69 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 + 26 T + 445 T^{2} + 4604 T^{3} + 445 p T^{4} + 26 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 398 T^{2} + 71991 T^{4} - 7610180 T^{6} + 71991 p^{2} T^{8} - 398 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 - 6 T + 15 T^{2} + 188 T^{3} + 15 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 - 402 T^{2} + 79331 T^{4} - 9565460 T^{6} + 79331 p^{2} T^{8} - 402 p^{4} T^{10} + p^{6} T^{12} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.49502115445240503263295553096, −4.30813300771414593835423957170, −4.09251266842098676810205321403, −4.05963809433703045354104343816, −3.84189747281471888938144246313, −3.59863614099654085249728311458, −3.55674168602339080079632327461, −3.35688173548907410371942501612, −3.28791847174755175056626734007, −3.04870458607400675070429381942, −2.89248786738415900704407369059, −2.69703934389160025444259154045, −2.50850159937470630845207701855, −2.47954697627113768009821596601, −2.46380480121972654620512450713, −1.99254294041878720120639405390, −1.64797676790851794209185882959, −1.51051979957863450251381268019, −1.46254025385839610955371610446, −1.42479544875468443842243438656, −1.38505100929604384259484377236, −1.28158527650333582157824441116, −0.67022896519519713265701968297, −0.26018709799994795815547576798, −0.25823291424071232560391514923,
0.25823291424071232560391514923, 0.26018709799994795815547576798, 0.67022896519519713265701968297, 1.28158527650333582157824441116, 1.38505100929604384259484377236, 1.42479544875468443842243438656, 1.46254025385839610955371610446, 1.51051979957863450251381268019, 1.64797676790851794209185882959, 1.99254294041878720120639405390, 2.46380480121972654620512450713, 2.47954697627113768009821596601, 2.50850159937470630845207701855, 2.69703934389160025444259154045, 2.89248786738415900704407369059, 3.04870458607400675070429381942, 3.28791847174755175056626734007, 3.35688173548907410371942501612, 3.55674168602339080079632327461, 3.59863614099654085249728311458, 3.84189747281471888938144246313, 4.05963809433703045354104343816, 4.09251266842098676810205321403, 4.30813300771414593835423957170, 4.49502115445240503263295553096
Plot not available for L-functions of degree greater than 10.
