| L(s) = 1 | − 4-s − 2·5-s − 9-s + 2·11-s + 16-s − 16·19-s + 2·20-s − 25-s + 10·29-s + 6·31-s + 36-s − 10·41-s − 2·44-s + 2·45-s − 11·49-s − 4·55-s + 8·59-s − 18·61-s − 64-s + 12·71-s + 16·76-s + 16·79-s − 2·80-s + 81-s + 8·89-s + 32·95-s − 2·99-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 0.894·5-s − 1/3·9-s + 0.603·11-s + 1/4·16-s − 3.67·19-s + 0.447·20-s − 1/5·25-s + 1.85·29-s + 1.07·31-s + 1/6·36-s − 1.56·41-s − 0.301·44-s + 0.298·45-s − 1.57·49-s − 0.539·55-s + 1.04·59-s − 2.30·61-s − 1/8·64-s + 1.42·71-s + 1.83·76-s + 1.80·79-s − 0.223·80-s + 1/9·81-s + 0.847·89-s + 3.28·95-s − 0.201·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5775030128\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5775030128\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 37 | $C_2$ | \( 1 + T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 81 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 113 T^{2} + p^{2} T^{4} \) |
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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
−10.53979398143835449083076821274, −9.395669002911796927189781335534, −9.357892467028456114938748667481, −8.570350274219051471220173676509, −8.327331849187970760981990550251, −8.246948449039429908119213937451, −7.84041266157207356084912538028, −6.99906153159766326917334432871, −6.52367819123424428092002382614, −6.43622346198207719935651175720, −6.05137938684773561815803510505, −5.09486671455846120428412360008, −4.80164103970649508307300584206, −4.34645260128868524479721101036, −3.96523033989564237250482058195, −3.53273829798565745141436587912, −2.78697484479330659016921360453, −2.23110081171226043300561723713, −1.46048655681344617088755378236, −0.34266992419124572471659020251,
0.34266992419124572471659020251, 1.46048655681344617088755378236, 2.23110081171226043300561723713, 2.78697484479330659016921360453, 3.53273829798565745141436587912, 3.96523033989564237250482058195, 4.34645260128868524479721101036, 4.80164103970649508307300584206, 5.09486671455846120428412360008, 6.05137938684773561815803510505, 6.43622346198207719935651175720, 6.52367819123424428092002382614, 6.99906153159766326917334432871, 7.84041266157207356084912538028, 8.246948449039429908119213937451, 8.327331849187970760981990550251, 8.570350274219051471220173676509, 9.357892467028456114938748667481, 9.395669002911796927189781335534, 10.53979398143835449083076821274
