| L(s) = 1 | − 2·4-s + 30·13-s + 4·16-s − 26·19-s + 48·25-s + 6·31-s + 34·37-s − 170·43-s − 60·52-s − 144·61-s − 8·64-s + 86·67-s − 190·73-s + 52·76-s + 138·79-s + 32·97-s − 96·100-s − 122·103-s − 130·109-s + 192·121-s − 12·124-s + 127-s + 131-s + 137-s + 139-s − 68·148-s + 149-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 2.30·13-s + 1/4·16-s − 1.36·19-s + 1.91·25-s + 6/31·31-s + 0.918·37-s − 3.95·43-s − 1.15·52-s − 2.36·61-s − 1/8·64-s + 1.28·67-s − 2.60·73-s + 0.684·76-s + 1.74·79-s + 0.329·97-s − 0.959·100-s − 1.18·103-s − 1.19·109-s + 1.58·121-s − 0.0967·124-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 0.459·148-s + 0.00671·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.035998630\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.035998630\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 48 T^{2} + p^{4} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 192 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 15 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 450 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 13 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 546 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 1170 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 3 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 17 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 3136 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 85 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 784 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 4466 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 1230 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 72 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 43 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 7344 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 95 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 69 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 10080 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 2590 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 16 T + p^{2} T^{2} )^{2} \) |
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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
−10.33483416973739514455699379715, −9.647689109089151303082928406159, −9.246404499572143332290508045652, −8.759400003704788231675182017771, −8.483446681638148964994144075885, −8.274869924013583988531479582796, −7.79454041246524013305767998474, −7.03183669353088332145770969657, −6.56689313010192626865148743191, −6.32571181011750943852912596866, −5.96909271287050820899040322624, −5.08831221549109123911256204597, −4.99037568207296011803907691133, −4.16355648055610461305929991480, −3.96546391951100975620875913794, −3.12239826401546163222934072125, −2.97975731402677073802145215034, −1.76748740565168009105784053164, −1.38706044404102170647020819007, −0.50210037649063395664577859525,
0.50210037649063395664577859525, 1.38706044404102170647020819007, 1.76748740565168009105784053164, 2.97975731402677073802145215034, 3.12239826401546163222934072125, 3.96546391951100975620875913794, 4.16355648055610461305929991480, 4.99037568207296011803907691133, 5.08831221549109123911256204597, 5.96909271287050820899040322624, 6.32571181011750943852912596866, 6.56689313010192626865148743191, 7.03183669353088332145770969657, 7.79454041246524013305767998474, 8.274869924013583988531479582796, 8.483446681638148964994144075885, 8.759400003704788231675182017771, 9.246404499572143332290508045652, 9.647689109089151303082928406159, 10.33483416973739514455699379715
