| L(s) = 1 | − 4·3-s + 6·9-s + 4·11-s − 16·33-s + 4·67-s − 15·81-s − 4·83-s − 4·97-s + 24·99-s + 8·121-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 − 16·201-s + ⋯ |
| L(s) = 1 | − 4·3-s + 6·9-s + 4·11-s − 16·33-s + 4·67-s − 15·81-s − 4·83-s − 4·97-s + 24·99-s + 8·121-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 − 16·201-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3088879393\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3088879393\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 13 | $C_2^2$ | \( 1 + T^{4} \) |
| good | 3 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 5 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 7 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 37 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \) |
| 71 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 83 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \) |
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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
−6.23419578781427181814755604812, −6.13750132679705129456962501401, −6.01322560834684572460328343623, −5.61994927179897205010427293936, −5.46646347498330350961052526113, −5.32776916692961839949951128195, −5.29032041527772161086589480255, −5.21978099986299939232585636297, −4.70825005235869427812366605484, −4.53230174590484081387741843906, −4.33137426971137953149802076577, −4.10589036630767346465375891242, −4.00832217688096604356692928227, −3.90134934504007201679611440278, −3.43724585778334133777786594958, −3.13049275423748141571399644721, −3.03683872640545921137125659944, −2.58238175967000925342056576699, −2.52894735991835085369705734373, −1.84555120008892771865255644502, −1.63651564532410532231685125454, −1.34855798225777769390683215577, −0.991848729997494632635025186410, −0.924966298934209022985765984865, −0.41622736699914386261097607715,
0.41622736699914386261097607715, 0.924966298934209022985765984865, 0.991848729997494632635025186410, 1.34855798225777769390683215577, 1.63651564532410532231685125454, 1.84555120008892771865255644502, 2.52894735991835085369705734373, 2.58238175967000925342056576699, 3.03683872640545921137125659944, 3.13049275423748141571399644721, 3.43724585778334133777786594958, 3.90134934504007201679611440278, 4.00832217688096604356692928227, 4.10589036630767346465375891242, 4.33137426971137953149802076577, 4.53230174590484081387741843906, 4.70825005235869427812366605484, 5.21978099986299939232585636297, 5.29032041527772161086589480255, 5.32776916692961839949951128195, 5.46646347498330350961052526113, 5.61994927179897205010427293936, 6.01322560834684572460328343623, 6.13750132679705129456962501401, 6.23419578781427181814755604812
