| L(s) = 1 | − 40·13-s + 64·25-s + 256·37-s + 124·49-s − 384·73-s + 256·97-s − 472·109-s − 156·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 324·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | − 3.07·13-s + 2.55·25-s + 6.91·37-s + 2.53·49-s − 5.26·73-s + 2.63·97-s − 4.33·109-s − 1.28·121-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 + 1.91·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(4.958799034\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.958799034\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( ( 1 - 62 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 78 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 10 T + p^{2} T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 + p^{4} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 82 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 738 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 1440 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 1762 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 64 T + p^{2} T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 3200 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 1138 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 4098 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 5280 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 61 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 3218 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 5598 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 96 T + p^{2} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 8482 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 1902 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 12800 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 64 T + p^{2} T^{2} )^{4} \) |
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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
−6.07997785502857837132459048223, −6.06729121048247341815992834340, −5.65362219212280282001158480086, −5.63233587419615620398526513732, −5.52109606883229986568312531088, −5.09126534152412285706058329412, −4.75907504955195506884049680827, −4.70626373644164298848242106178, −4.52702071325911836180804265827, −4.47687918030992563116606367490, −4.07086866728359504860605385021, −4.01679505943096161871087044185, −3.76945183239463906161068365013, −3.06845731539166771566802419294, −2.86563455356268276913522033211, −2.80196678967592136093360660724, −2.75877845965032638672789564542, −2.56343518950529630519506787222, −2.24641112648061256270004100247, −1.86927404992168345405295470684, −1.52040837270404785138275962152, −1.08215031459177405957993223740, −0.842914089828806321312132988969, −0.60174735108095354493928060323, −0.31725035254941677784755141019,
0.31725035254941677784755141019, 0.60174735108095354493928060323, 0.842914089828806321312132988969, 1.08215031459177405957993223740, 1.52040837270404785138275962152, 1.86927404992168345405295470684, 2.24641112648061256270004100247, 2.56343518950529630519506787222, 2.75877845965032638672789564542, 2.80196678967592136093360660724, 2.86563455356268276913522033211, 3.06845731539166771566802419294, 3.76945183239463906161068365013, 4.01679505943096161871087044185, 4.07086866728359504860605385021, 4.47687918030992563116606367490, 4.52702071325911836180804265827, 4.70626373644164298848242106178, 4.75907504955195506884049680827, 5.09126534152412285706058329412, 5.52109606883229986568312531088, 5.63233587419615620398526513732, 5.65362219212280282001158480086, 6.06729121048247341815992834340, 6.07997785502857837132459048223
