| L(s) = 1 | + 2·4-s + 9-s + 16-s − 2·19-s − 4·31-s + 2·36-s + 49-s + 2·61-s − 2·64-s − 4·76-s − 2·79-s − 2·109-s − 2·121-s − 8·124-s + 127-s + 131-s + 137-s + 139-s + 144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s − 2·171-s + 173-s + ⋯ |
| L(s) = 1 | + 2·4-s + 9-s + 16-s − 2·19-s − 4·31-s + 2·36-s + 49-s + 2·61-s − 2·64-s − 4·76-s − 2·79-s − 2·109-s − 2·121-s − 8·124-s + 127-s + 131-s + 137-s + 139-s + 144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s − 2·171-s + 173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9002772190\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9002772190\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| good | 2 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 31 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 37 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 73 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
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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
−8.070664119349607603211823988416, −7.67425609426367665189068310467, −7.52305118607046320499038417515, −7.17021858082527622158894999152, −7.14948455464147816042188569858, −6.78204462194694503729236379369, −6.71585031807982118020629669133, −6.67304428559958013983450015879, −6.21648645733122950963363322555, −5.82541780777366235135103980737, −5.67969635074650773342840217999, −5.40749832722790060450139866586, −5.38568501068442250150551148058, −4.72499988851361415917537063770, −4.37968090057031358885415286030, −4.29285374354381864229391367999, −3.92913329207248560771198678931, −3.80054616729598062928015936041, −3.14017202034808787869760428142, −3.12563581516482922057166614727, −2.57763652978907088229204792629, −2.15564969205768814216054890916, −1.94263208975749447212563756240, −1.88230425332256657895875120726, −1.31700970631783550491257665869,
1.31700970631783550491257665869, 1.88230425332256657895875120726, 1.94263208975749447212563756240, 2.15564969205768814216054890916, 2.57763652978907088229204792629, 3.12563581516482922057166614727, 3.14017202034808787869760428142, 3.80054616729598062928015936041, 3.92913329207248560771198678931, 4.29285374354381864229391367999, 4.37968090057031358885415286030, 4.72499988851361415917537063770, 5.38568501068442250150551148058, 5.40749832722790060450139866586, 5.67969635074650773342840217999, 5.82541780777366235135103980737, 6.21648645733122950963363322555, 6.67304428559958013983450015879, 6.71585031807982118020629669133, 6.78204462194694503729236379369, 7.14948455464147816042188569858, 7.17021858082527622158894999152, 7.52305118607046320499038417515, 7.67425609426367665189068310467, 8.070664119349607603211823988416
