| L(s) = 1 | + 96·25-s − 196·49-s + 384·73-s + 576·97-s − 484·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 476·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
| L(s) = 1 | + 3.83·25-s − 4·49-s + 5.26·73-s + 5.93·97-s − 4·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 2.81·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + 0.00448·223-s + 0.00440·227-s + 0.00436·229-s + 0.00429·233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.575559598\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.575559598\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 - 48 T^{2} + p^{4} T^{4} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 11 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 - 24 T + p^{2} T^{2} )^{2}( 1 + 24 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 480 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 - 1680 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 - 70 T + p^{2} T^{2} )^{2}( 1 + 70 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 1440 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 + 5040 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 - 22 T + p^{2} T^{2} )^{2}( 1 + 22 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 73 | $C_2$ | \( ( 1 - 96 T + p^{2} T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 83 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 89 | $C_2^2$ | \( ( 1 + 12480 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 144 T + p^{2} T^{2} )^{4} \) |
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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
−6.72636004993601581078439126078, −6.65192013682059792633711609475, −6.46810751581320529497343579966, −6.16491310470937410989795935132, −6.12289154755751810997939994481, −5.60225501004370182483661787596, −5.31411277635712928342689350984, −5.21407872125335133258923807327, −4.92712922070975205200476765346, −4.74847545848867814154751801821, −4.60662634284877627184308172637, −4.55328357602598938318646759763, −3.83779762637665749267530585829, −3.65265284661471654183532101022, −3.57484124200022162205680399235, −3.17754740210041464340409088623, −2.99364016217899380456399946920, −2.82612526756517718307947603279, −2.22137680561867554141745221398, −2.15576274935801890407235415075, −1.88411263827581358936132554817, −1.20259986135323161180100332588, −1.06584785149749899323945464687, −0.815554670027512344151957023280, −0.23283083058212024976003199615,
0.23283083058212024976003199615, 0.815554670027512344151957023280, 1.06584785149749899323945464687, 1.20259986135323161180100332588, 1.88411263827581358936132554817, 2.15576274935801890407235415075, 2.22137680561867554141745221398, 2.82612526756517718307947603279, 2.99364016217899380456399946920, 3.17754740210041464340409088623, 3.57484124200022162205680399235, 3.65265284661471654183532101022, 3.83779762637665749267530585829, 4.55328357602598938318646759763, 4.60662634284877627184308172637, 4.74847545848867814154751801821, 4.92712922070975205200476765346, 5.21407872125335133258923807327, 5.31411277635712928342689350984, 5.60225501004370182483661787596, 6.12289154755751810997939994481, 6.16491310470937410989795935132, 6.46810751581320529497343579966, 6.65192013682059792633711609475, 6.72636004993601581078439126078
