| L(s) = 1 | + 16·11-s − 64·25-s − 52·49-s + 320·59-s − 224·73-s + 304·83-s + 416·97-s + 512·107-s − 324·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 28·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | + 1.45·11-s − 2.55·25-s − 1.06·49-s + 5.42·59-s − 3.06·73-s + 3.66·83-s + 4.28·97-s + 4.78·107-s − 2.67·121-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 + 0.165·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(5.383492406\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.383492406\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( ( 1 + 32 T^{2} + p^{4} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p^{2} T^{2} )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 560 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 34 T + p^{2} T^{2} )^{2}( 1 + 34 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 238 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 1520 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 1850 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 1442 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 2480 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 910 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 3122 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 880 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 80 T + p^{2} T^{2} )^{4} \) |
| 61 | $C_2^2$ | \( ( 1 - 6146 T^{2} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 62 T + p^{2} T^{2} )^{2}( 1 + 62 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 1582 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 56 T + p^{2} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 11834 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 76 T + p^{2} T^{2} )^{4} \) |
| 89 | $C_2^2$ | \( ( 1 - 7904 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 104 T + p^{2} T^{2} )^{4} \) |
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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
−6.95571019548950905130633028107, −6.36401765663643260902450700004, −6.30835572071227202635511446185, −6.27081626659320399115143260997, −6.03715155375870151585396668912, −5.64729178007761870323325434888, −5.52623766330207826741934795075, −5.16312554061074105218085385110, −5.16284948128031563052929117711, −4.58647272851736845363693396776, −4.49258734848862739914239470963, −4.31957144254675947273876415785, −3.97168226338377604423732315746, −3.65203925398221482751252996849, −3.53726560943620849882192016433, −3.47449546861867262173630192698, −3.01603773321547492804364745141, −2.66820193789950491497642490374, −2.08341261330756040288038127505, −2.02300587389276465915435975041, −1.99443565508377841565234601681, −1.51569717919210617641285910935, −0.805207367644638355913491045223, −0.803803015436518391410579244605, −0.38130728394347424710187129916,
0.38130728394347424710187129916, 0.803803015436518391410579244605, 0.805207367644638355913491045223, 1.51569717919210617641285910935, 1.99443565508377841565234601681, 2.02300587389276465915435975041, 2.08341261330756040288038127505, 2.66820193789950491497642490374, 3.01603773321547492804364745141, 3.47449546861867262173630192698, 3.53726560943620849882192016433, 3.65203925398221482751252996849, 3.97168226338377604423732315746, 4.31957144254675947273876415785, 4.49258734848862739914239470963, 4.58647272851736845363693396776, 5.16284948128031563052929117711, 5.16312554061074105218085385110, 5.52623766330207826741934795075, 5.64729178007761870323325434888, 6.03715155375870151585396668912, 6.27081626659320399115143260997, 6.30835572071227202635511446185, 6.36401765663643260902450700004, 6.95571019548950905130633028107
