| L(s) = 1 | − 4-s − 9-s − 8·11-s + 16-s − 8·19-s + 4·29-s − 2·31-s + 36-s + 20·41-s + 8·44-s − 2·49-s − 16·59-s + 20·61-s − 64-s + 8·76-s + 16·79-s + 81-s + 12·89-s + 8·99-s + 36·101-s − 12·109-s − 4·116-s + 26·121-s + 2·124-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1/3·9-s − 2.41·11-s + 1/4·16-s − 1.83·19-s + 0.742·29-s − 0.359·31-s + 1/6·36-s + 3.12·41-s + 1.20·44-s − 2/7·49-s − 2.08·59-s + 2.56·61-s − 1/8·64-s + 0.917·76-s + 1.80·79-s + 1/9·81-s + 1.27·89-s + 0.804·99-s + 3.58·101-s − 1.14·109-s − 0.371·116-s + 2.36·121-s + 0.179·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21622500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21622500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.168495665\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.168495665\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 31 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 158 T^{2} + p^{2} T^{4} \) |
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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
−8.394022292481085227518729019548, −8.022034066296966564146756463756, −7.75475318147133919743262546715, −7.73997882823514993689838467110, −7.12948124301263205297912561495, −6.51048945565580877858949820364, −6.36120002142041072559448162399, −5.89798258113482087571171974761, −5.51165096256350137043810090187, −5.15851472978656081536133065093, −4.85697016481569259983874591109, −4.35348170402238053517335743892, −4.15747629675926135435545382868, −3.52132930027113168396251459676, −3.06142356309532815642618560495, −2.53050988853198620136101480404, −2.34288383281389928116130857484, −1.86814498157576651718195197531, −0.812648249560500697296726980955, −0.38996351335477283426337970550,
0.38996351335477283426337970550, 0.812648249560500697296726980955, 1.86814498157576651718195197531, 2.34288383281389928116130857484, 2.53050988853198620136101480404, 3.06142356309532815642618560495, 3.52132930027113168396251459676, 4.15747629675926135435545382868, 4.35348170402238053517335743892, 4.85697016481569259983874591109, 5.15851472978656081536133065093, 5.51165096256350137043810090187, 5.89798258113482087571171974761, 6.36120002142041072559448162399, 6.51048945565580877858949820364, 7.12948124301263205297912561495, 7.73997882823514993689838467110, 7.75475318147133919743262546715, 8.022034066296966564146756463756, 8.394022292481085227518729019548
