| L(s) = 1 | + 6·5-s + 8·13-s + 48·17-s − 23·25-s + 84·29-s − 20·37-s + 108·41-s + 95·49-s + 90·53-s − 32·61-s + 48·65-s + 2·73-s + 288·85-s − 156·89-s − 274·97-s − 126·101-s − 148·109-s − 132·113-s + 215·121-s − 342·125-s + 127-s + 131-s + 137-s + 139-s + 504·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 6/5·5-s + 8/13·13-s + 2.82·17-s − 0.919·25-s + 2.89·29-s − 0.540·37-s + 2.63·41-s + 1.93·49-s + 1.69·53-s − 0.524·61-s + 0.738·65-s + 2/73·73-s + 3.38·85-s − 1.75·89-s − 2.82·97-s − 1.24·101-s − 1.35·109-s − 1.16·113-s + 1.77·121-s − 2.73·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 3.47·145-s + 0.00671·149-s + 0.00662·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 186624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 186624 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.872877783\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.872877783\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p^{2} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 95 T^{2} + p^{4} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 215 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 24 T + p^{2} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 710 T^{2} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 86 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 42 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 599 T^{2} + p^{4} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 54 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 998 T^{2} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 1718 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 45 T + p^{2} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 6530 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 16 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 1114 T^{2} + p^{4} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 3170 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 1390 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 4031 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 78 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 137 T + p^{2} T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.00883630559951699640030890662, −10.55673986337658114337735695326, −10.21104690721741663381870625407, −9.859141160421757795199451149985, −9.517316069564264573648372299469, −9.006315247814377519332180419345, −8.230081058688135007005617404587, −8.202277437963871792096662147617, −7.40815718493043206818293602440, −7.08392663420239517128157012831, −6.11172402280219941578981654312, −6.09247783383913637599149199360, −5.45718078487645683803065159966, −5.20501802092991997688149286573, −4.11298203618883809786376492872, −3.87233765480566508411203567847, −2.77284063133093171083284616997, −2.60695187354135028240518397640, −1.36107667658714808538922852467, −0.992439609253175337248362234172,
0.992439609253175337248362234172, 1.36107667658714808538922852467, 2.60695187354135028240518397640, 2.77284063133093171083284616997, 3.87233765480566508411203567847, 4.11298203618883809786376492872, 5.20501802092991997688149286573, 5.45718078487645683803065159966, 6.09247783383913637599149199360, 6.11172402280219941578981654312, 7.08392663420239517128157012831, 7.40815718493043206818293602440, 8.202277437963871792096662147617, 8.230081058688135007005617404587, 9.006315247814377519332180419345, 9.517316069564264573648372299469, 9.859141160421757795199451149985, 10.21104690721741663381870625407, 10.55673986337658114337735695326, 11.00883630559951699640030890662
