| L(s) = 1 | + 4·5-s + 2·9-s + 10·25-s − 8·29-s + 8·45-s + 20·49-s − 40·53-s + 40·73-s − 5·81-s − 56·97-s − 72·101-s + 36·121-s + 20·125-s + 127-s + 131-s + 137-s + 139-s − 32·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 36·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 2/3·9-s + 2·25-s − 1.48·29-s + 1.19·45-s + 20/7·49-s − 5.49·53-s + 4.68·73-s − 5/9·81-s − 5.68·97-s − 7.16·101-s + 3.27·121-s + 1.78·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2.65·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2.76·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.030257180\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.030257180\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 7 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 66 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 126 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 154 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 162 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2}( 1 + 18 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{4} \) |
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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
−7.03874145813198253572189832688, −6.98577340369585245011517830498, −6.75317446075505545598636591550, −6.57698658021947336667412588707, −6.25484136016154671241610743164, −6.05365236704014094228295004864, −5.68627750473301771784588732821, −5.55446343215885711025605926362, −5.54201223340379940156585743444, −5.24880537936223523427300262249, −4.79649688976664199970180325729, −4.73114161479892060249917934667, −4.35444004662881801739351583015, −4.21931667394107737464662669414, −3.79931467082068675718163889788, −3.53868158663068580108511800570, −3.37000754329997649828098296909, −2.88159624100861602716160471689, −2.55123815164734002069201417128, −2.48108612847455312881335281237, −2.08786402445296695303455307735, −1.60185406474535819252373424464, −1.40816925304107073261244093385, −1.27478694381970216509052049936, −0.36775195076258271793733122621,
0.36775195076258271793733122621, 1.27478694381970216509052049936, 1.40816925304107073261244093385, 1.60185406474535819252373424464, 2.08786402445296695303455307735, 2.48108612847455312881335281237, 2.55123815164734002069201417128, 2.88159624100861602716160471689, 3.37000754329997649828098296909, 3.53868158663068580108511800570, 3.79931467082068675718163889788, 4.21931667394107737464662669414, 4.35444004662881801739351583015, 4.73114161479892060249917934667, 4.79649688976664199970180325729, 5.24880537936223523427300262249, 5.54201223340379940156585743444, 5.55446343215885711025605926362, 5.68627750473301771784588732821, 6.05365236704014094228295004864, 6.25484136016154671241610743164, 6.57698658021947336667412588707, 6.75317446075505545598636591550, 6.98577340369585245011517830498, 7.03874145813198253572189832688
