| L(s) = 1 | + 12·13-s − 16-s + 16·19-s − 8·25-s − 12·37-s − 32·43-s − 8·61-s − 36·73-s + 64·79-s − 4·97-s + 16·103-s − 20·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 82·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 3.32·13-s − 1/4·16-s + 3.67·19-s − 8/5·25-s − 1.97·37-s − 4.87·43-s − 1.02·61-s − 4.21·73-s + 7.20·79-s − 0.406·97-s + 1.57·103-s − 1.91·109-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 + 6.30·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.598474259\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.598474259\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + T^{4} \) |
| 3 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| good | 7 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 32 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 40 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 56 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - 4046 T^{4} + p^{4} T^{8} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^3$ | \( 1 + 5794 T^{4} + p^{4} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{4} \) |
| 83 | $C_2^3$ | \( 1 + 8722 T^{4} + p^{4} T^{8} \) |
| 89 | $C_2^3$ | \( 1 - 15518 T^{4} + p^{4} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.91299351532637768288188242905, −6.80283901529697358486118445062, −6.63215696155212453514007099102, −6.12366772622573330064621939201, −6.04045312559177259296006463979, −6.01379649692109139475835492563, −5.83928301931656073563105830684, −5.18753732371724558055023104424, −5.15077760293663593620197948421, −5.11072879495428417132829126567, −5.01126847443075777366080717253, −4.42695048070757368741357597826, −4.07939156550319562784191123665, −3.89823075083512290365013791722, −3.66272953494646273664356568108, −3.26948047967258636595516237607, −3.24655572632053368147659495095, −3.18730023546970179537677952456, −2.93715794696060300502930368511, −2.04555647302844434451047677508, −1.73438810106848116807734246571, −1.72937902140355706449893069152, −1.44876358567937925324216587932, −0.839873750259314010956275594183, −0.53963126678503738045683839721,
0.53963126678503738045683839721, 0.839873750259314010956275594183, 1.44876358567937925324216587932, 1.72937902140355706449893069152, 1.73438810106848116807734246571, 2.04555647302844434451047677508, 2.93715794696060300502930368511, 3.18730023546970179537677952456, 3.24655572632053368147659495095, 3.26948047967258636595516237607, 3.66272953494646273664356568108, 3.89823075083512290365013791722, 4.07939156550319562784191123665, 4.42695048070757368741357597826, 5.01126847443075777366080717253, 5.11072879495428417132829126567, 5.15077760293663593620197948421, 5.18753732371724558055023104424, 5.83928301931656073563105830684, 6.01379649692109139475835492563, 6.04045312559177259296006463979, 6.12366772622573330064621939201, 6.63215696155212453514007099102, 6.80283901529697358486118445062, 6.91299351532637768288188242905
