| L(s) = 1 | − 16·4-s + 342·9-s − 242·11-s + 256·16-s + 1.90e3·19-s + 124·29-s − 1.51e4·31-s − 5.47e3·36-s − 1.36e4·41-s + 3.87e3·44-s + 3.06e4·49-s − 9.65e4·59-s + 3.50e4·61-s − 4.09e3·64-s − 4.58e4·71-s − 3.04e4·76-s − 1.04e5·79-s + 5.79e4·81-s − 8.39e4·89-s − 8.27e4·99-s + 1.14e5·101-s − 1.19e5·109-s − 1.98e3·116-s + 4.39e4·121-s + 2.41e5·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 1.40·9-s − 0.603·11-s + 1/4·16-s + 1.20·19-s + 0.0273·29-s − 2.82·31-s − 0.703·36-s − 1.26·41-s + 0.301·44-s + 1.82·49-s − 3.61·59-s + 1.20·61-s − 1/8·64-s − 1.07·71-s − 0.604·76-s − 1.88·79-s + 0.980·81-s − 1.12·89-s − 0.848·99-s + 1.11·101-s − 0.967·109-s − 0.0136·116-s + 3/11·121-s + 1.41·124-s + 5.50e−6·127-s + 5.09e−6·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 302500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 302500 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.7468414363\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7468414363\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_2$ | \( 1 + p^{4} T^{2} \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 38 p^{2} T^{2} + p^{10} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 30698 T^{2} + p^{10} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 450986 T^{2} + p^{10} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 9426 p^{2} T^{2} + p^{10} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 952 T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 11680222 T^{2} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 62 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 7560 T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 54305318 T^{2} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6818 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 116860786 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 43966386 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 450111270 T^{2} + p^{10} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 48292 T + p^{5} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 17530 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 1451051878 T^{2} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 22912 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 1856329282 T^{2} + p^{10} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 52396 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 7815829186 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 41958 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 15760770110 T^{2} + p^{10} T^{4} \) |
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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
−10.31253500270475538265323791018, −9.658307444966813614261738359810, −9.520673380735974292840527445597, −8.781620443634072984168757903124, −8.734494200959694695819037840341, −7.69920203828220826373517171852, −7.66516076813153424817697957394, −7.10580630192726482356162654192, −6.87578070280878620209866756323, −5.79142373792431701101718990507, −5.77306972522642993373537410718, −5.01821732339159486779285812883, −4.66937020943132837263732695714, −4.05691728203034890120506054659, −3.56038676407367105927459295097, −3.08176624041432816463495932290, −2.24768482235258438722640924969, −1.52190239609972451170481917971, −1.21575568304057233367920266054, −0.20254960858457299953326350271,
0.20254960858457299953326350271, 1.21575568304057233367920266054, 1.52190239609972451170481917971, 2.24768482235258438722640924969, 3.08176624041432816463495932290, 3.56038676407367105927459295097, 4.05691728203034890120506054659, 4.66937020943132837263732695714, 5.01821732339159486779285812883, 5.77306972522642993373537410718, 5.79142373792431701101718990507, 6.87578070280878620209866756323, 7.10580630192726482356162654192, 7.66516076813153424817697957394, 7.69920203828220826373517171852, 8.734494200959694695819037840341, 8.781620443634072984168757903124, 9.520673380735974292840527445597, 9.658307444966813614261738359810, 10.31253500270475538265323791018
