| L(s) = 1 | + 2·4-s + 12·9-s + 16·11-s − 5·16-s + 8·19-s − 8·29-s − 16·31-s + 24·36-s + 4·41-s + 32·44-s − 36·61-s − 20·64-s + 16·76-s + 8·79-s + 90·81-s + 28·89-s + 192·99-s − 4·101-s + 40·109-s − 16·116-s + 128·121-s − 32·124-s + 127-s + 131-s + 137-s + 139-s − 60·144-s + ⋯ |
| L(s) = 1 | + 4-s + 4·9-s + 4.82·11-s − 5/4·16-s + 1.83·19-s − 1.48·29-s − 2.87·31-s + 4·36-s + 0.624·41-s + 4.82·44-s − 4.60·61-s − 5/2·64-s + 1.83·76-s + 0.900·79-s + 10·81-s + 2.96·89-s + 19.2·99-s − 0.398·101-s + 3.83·109-s − 1.48·116-s + 11.6·121-s − 2.87·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 5·144-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.988497590\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.988497590\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| good | 2 | $C_2^2$ | \( ( 1 - T^{2} + p^{2} T^{4} )^{2} \) |
| 3 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 7 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \) |
| 11 | $C_2^2$ | \( ( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^3$ | \( 1 - 142 T^{4} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 238 T^{4} + p^{4} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 54 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 66 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^3$ | \( 1 - 2702 T^{4} + p^{4} T^{8} \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 61 | $C_2^2$ | \( ( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 6398 T^{4} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 126 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 22 T + 242 T^{2} - 22 p T^{3} + p^{2} T^{4} )( 1 + 22 T + 242 T^{2} + 22 p T^{3} + p^{2} T^{4} ) \) |
| 89 | $C_2^2$ | \( ( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 174 T^{2} + p^{2} T^{4} )^{2} \) |
| 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
−7.36810030931368005630144869504, −7.29900167194767675144871450527, −7.19106774701321637461178132177, −6.71385539111449231587168571173, −6.63513986018119108592325641281, −6.45670073165580359800912194663, −6.11531943848649549094240119005, −6.06227702888113593816081160643, −5.91790464333501754952404888116, −5.14550684794159414929233760858, −4.93474394528426623314835201522, −4.78580527348481998391933332409, −4.50129014447432163138427666023, −4.12150757926765518937995571478, −3.96434248664853086030964984420, −3.95760664875555329702150104354, −3.57290454014270065124473920355, −3.26401193636293725225566517783, −3.20339209386446897602257084474, −2.06555761095676407786282457783, −1.98770239337389398298616446257, −1.84945081486006008176496898445, −1.45656544434698960730519338340, −1.24785424071002230282571312133, −0.942753274648310271344165429892,
0.942753274648310271344165429892, 1.24785424071002230282571312133, 1.45656544434698960730519338340, 1.84945081486006008176496898445, 1.98770239337389398298616446257, 2.06555761095676407786282457783, 3.20339209386446897602257084474, 3.26401193636293725225566517783, 3.57290454014270065124473920355, 3.95760664875555329702150104354, 3.96434248664853086030964984420, 4.12150757926765518937995571478, 4.50129014447432163138427666023, 4.78580527348481998391933332409, 4.93474394528426623314835201522, 5.14550684794159414929233760858, 5.91790464333501754952404888116, 6.06227702888113593816081160643, 6.11531943848649549094240119005, 6.45670073165580359800912194663, 6.63513986018119108592325641281, 6.71385539111449231587168571173, 7.19106774701321637461178132177, 7.29900167194767675144871450527, 7.36810030931368005630144869504
