| L(s) = 1 | − 3·9-s + 8·11-s − 20·19-s + 20·31-s + 16·41-s + 8·49-s + 20·61-s − 16·71-s − 28·79-s + 6·81-s − 4·89-s − 24·99-s + 12·109-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 38·169-s + 60·171-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | − 9-s + 2.41·11-s − 4.58·19-s + 3.59·31-s + 2.49·41-s + 8/7·49-s + 2.56·61-s − 1.89·71-s − 3.15·79-s + 2/3·81-s − 0.423·89-s − 2.41·99-s + 1.14·109-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 + 2.92·169-s + 4.58·171-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{6} \cdot 5^{12} \cdot 23^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{6} \cdot 5^{12} \cdot 23^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.814961841\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.814961841\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 + T^{2} )^{3} \) |
| 5 | \( 1 \) |
| 23 | \( ( 1 + T^{2} )^{3} \) |
| good | 7 | \( 1 - 8 T^{2} + 104 T^{4} - 782 T^{6} + 104 p^{2} T^{8} - 8 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( ( 1 - 4 T + 19 T^{2} - 40 T^{3} + 19 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 13 | \( ( 1 - 4 T - 11 T^{2} + 112 T^{3} - 11 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )( 1 + 4 T - 11 T^{2} - 112 T^{3} - 11 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} ) \) |
| 17 | \( 1 - 60 T^{2} + 1920 T^{4} - 40102 T^{6} + 1920 p^{2} T^{8} - 60 p^{4} T^{10} + p^{6} T^{12} \) |
| 19 | \( ( 1 + 10 T + 71 T^{2} + 328 T^{3} + 71 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 29 | \( ( 1 + 66 T^{2} + 18 T^{3} + 66 p T^{4} + p^{3} T^{6} )^{2} \) |
| 31 | \( ( 1 - 10 T + 110 T^{2} - 616 T^{3} + 110 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( 1 - 92 T^{2} + 4832 T^{4} - 195314 T^{6} + 4832 p^{2} T^{8} - 92 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( ( 1 - 8 T + 22 T^{2} - 74 T^{3} + 22 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( 1 - 42 T^{2} + 711 T^{4} + 27572 T^{6} + 711 p^{2} T^{8} - 42 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( 1 - 166 T^{2} + 15555 T^{4} - 889772 T^{6} + 15555 p^{2} T^{8} - 166 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( 1 - 52 T^{2} + 5256 T^{4} - 224030 T^{6} + 5256 p^{2} T^{8} - 52 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( ( 1 + 156 T^{2} - 18 T^{3} + 156 p T^{4} + p^{3} T^{6} )^{2} \) |
| 61 | \( ( 1 - 10 T + 161 T^{2} - 928 T^{3} + 161 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 - 336 T^{2} + 50016 T^{4} - 4287046 T^{6} + 50016 p^{2} T^{8} - 336 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( ( 1 + 8 T + 112 T^{2} + 554 T^{3} + 112 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( ( 1 - 24 T + 297 T^{2} - 2640 T^{3} + 297 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} )( 1 + 24 T + 297 T^{2} + 2640 T^{3} + 297 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} ) \) |
| 79 | \( ( 1 + 14 T + 3 p T^{2} + 2116 T^{3} + 3 p^{2} T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 364 T^{2} + 63696 T^{4} - 6649934 T^{6} + 63696 p^{2} T^{8} - 364 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 + 2 T + 67 T^{2} - 556 T^{3} + 67 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 - 6 T^{2} + 18831 T^{4} + 91244 T^{6} + 18831 p^{2} T^{8} - 6 p^{4} T^{10} + p^{6} T^{12} \) |
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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
−4.08839807211199519017279939242, −4.03439670222630577315164458944, −3.97890220102551953194422674026, −3.79638087354649584403803918501, −3.45472471047459043303472096004, −3.44065134917038690036061163410, −3.12952953314253894517817089065, −2.97620399042159755398151797308, −2.96445436767172907508465024152, −2.89851172526527792771628969114, −2.63108951382426878518205432549, −2.34743381211373651102793988624, −2.30160904689312080271943974727, −2.29491963370830031195034529787, −2.24155923545212858575972145437, −1.99553792603682464508351726376, −1.71898196928071942447854102984, −1.46312776436610417168769319279, −1.40903600137974720641253581772, −1.08893143056791983158340530130, −1.07367719438634983631719053412, −0.989565487891282123730398187473, −0.52383143579804121441852865610, −0.33635697090319372187821031634, −0.24385565471411162343199355257,
0.24385565471411162343199355257, 0.33635697090319372187821031634, 0.52383143579804121441852865610, 0.989565487891282123730398187473, 1.07367719438634983631719053412, 1.08893143056791983158340530130, 1.40903600137974720641253581772, 1.46312776436610417168769319279, 1.71898196928071942447854102984, 1.99553792603682464508351726376, 2.24155923545212858575972145437, 2.29491963370830031195034529787, 2.30160904689312080271943974727, 2.34743381211373651102793988624, 2.63108951382426878518205432549, 2.89851172526527792771628969114, 2.96445436767172907508465024152, 2.97620399042159755398151797308, 3.12952953314253894517817089065, 3.44065134917038690036061163410, 3.45472471047459043303472096004, 3.79638087354649584403803918501, 3.97890220102551953194422674026, 4.03439670222630577315164458944, 4.08839807211199519017279939242
Plot not available for L-functions of degree greater than 10.
