| L(s) = 1 | − 3·8-s + 30·25-s − 42·49-s + 64-s + 36·101-s − 66·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s − 90·200-s + 211-s + 223-s + 227-s + 229-s + ⋯ |
| L(s) = 1 | − 1.06·8-s + 6·25-s − 6·49-s + 1/8·64-s + 3.58·101-s − 6·121-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 + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s − 6.36·200-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 23^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 23^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7919629453\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7919629453\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 3 T^{3} + p^{3} T^{6} \) |
| 23 | \( ( 1 + p T^{2} )^{3} \) |
| good | 3 | \( ( 1 - 4 T^{3} + p^{3} T^{6} )( 1 + 4 T^{3} + p^{3} T^{6} ) \) |
| 5 | \( ( 1 - p T^{2} )^{6} \) |
| 7 | \( ( 1 + p T^{2} )^{6} \) |
| 11 | \( ( 1 + p T^{2} )^{6} \) |
| 13 | \( ( 1 + 74 T^{3} + p^{3} T^{6} )^{2} \) |
| 17 | \( ( 1 - p T^{2} )^{6} \) |
| 19 | \( ( 1 + p T^{2} )^{6} \) |
| 29 | \( ( 1 + 282 T^{3} + p^{3} T^{6} )^{2} \) |
| 31 | \( ( 1 - 344 T^{3} + p^{3} T^{6} )( 1 + 344 T^{3} + p^{3} T^{6} ) \) |
| 37 | \( ( 1 - p T^{2} )^{6} \) |
| 41 | \( ( 1 - 426 T^{3} + p^{3} T^{6} )^{2} \) |
| 43 | \( ( 1 + p T^{2} )^{6} \) |
| 47 | \( ( 1 - 48 T^{3} + p^{3} T^{6} )( 1 + 48 T^{3} + p^{3} T^{6} ) \) |
| 53 | \( ( 1 - p T^{2} )^{6} \) |
| 59 | \( ( 1 - 12 T + p T^{2} )^{3}( 1 + 12 T + p T^{2} )^{3} \) |
| 61 | \( ( 1 - p T^{2} )^{6} \) |
| 67 | \( ( 1 + p T^{2} )^{6} \) |
| 71 | \( ( 1 - 1176 T^{3} + p^{3} T^{6} )( 1 + 1176 T^{3} + p^{3} T^{6} ) \) |
| 73 | \( ( 1 - 1226 T^{3} + p^{3} T^{6} )^{2} \) |
| 79 | \( ( 1 + p T^{2} )^{6} \) |
| 83 | \( ( 1 + p T^{2} )^{6} \) |
| 89 | \( ( 1 - p T^{2} )^{6} \) |
| 97 | \( ( 1 - p T^{2} )^{6} \) |
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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
−8.189106001821206401409176061943, −7.85595665357530602872889625750, −7.57897523630182724451143790359, −7.28179590331530861407703700313, −6.95118490259484901301505413767, −6.93882924848860774567587582968, −6.84349211583957573690627773210, −6.27107276935781090663118539355, −6.24743141901396155617083248304, −6.17412101188377704213868847961, −6.12007961118409824285742673463, −5.24670831538095665782551609687, −5.17249955999227184454121311703, −5.01967028742978629934617855274, −4.89918628465651346600288367418, −4.67377803414451380113431098376, −4.42220868242180061548484932379, −3.74303582627021454666040648183, −3.55950881173441346025792719267, −3.33978666997632839420065090311, −2.83912621843438152378424633431, −2.83392605673296513729713089253, −2.56747103653534940794669672036, −1.62670093299634777105663758010, −1.21001969050650181210683268297,
1.21001969050650181210683268297, 1.62670093299634777105663758010, 2.56747103653534940794669672036, 2.83392605673296513729713089253, 2.83912621843438152378424633431, 3.33978666997632839420065090311, 3.55950881173441346025792719267, 3.74303582627021454666040648183, 4.42220868242180061548484932379, 4.67377803414451380113431098376, 4.89918628465651346600288367418, 5.01967028742978629934617855274, 5.17249955999227184454121311703, 5.24670831538095665782551609687, 6.12007961118409824285742673463, 6.17412101188377704213868847961, 6.24743141901396155617083248304, 6.27107276935781090663118539355, 6.84349211583957573690627773210, 6.93882924848860774567587582968, 6.95118490259484901301505413767, 7.28179590331530861407703700313, 7.57897523630182724451143790359, 7.85595665357530602872889625750, 8.189106001821206401409176061943
Plot not available for L-functions of degree greater than 10.
