| L(s) = 1 | + 4·7-s − 9-s − 4·17-s − 8·23-s + 6·25-s − 4·31-s + 4·41-s + 24·47-s − 2·49-s − 4·63-s − 24·71-s − 12·73-s − 20·79-s + 81-s − 20·89-s − 4·97-s + 12·103-s − 12·113-s − 16·119-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 4·153-s + ⋯ |
| L(s) = 1 | + 1.51·7-s − 1/3·9-s − 0.970·17-s − 1.66·23-s + 6/5·25-s − 0.718·31-s + 0.624·41-s + 3.50·47-s − 2/7·49-s − 0.503·63-s − 2.84·71-s − 1.40·73-s − 2.25·79-s + 1/9·81-s − 2.11·89-s − 0.406·97-s + 1.18·103-s − 1.12·113-s − 1.46·119-s + 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.323·153-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.056662136\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.056662136\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−14.30393638177530778992496047296, −13.87107562266462274065499943418, −13.20381345458436835000960906529, −12.66531071422897734789510686624, −11.93773832768940573178041956869, −11.69428189996392567235340776374, −10.85374953070680384436830654252, −10.80976057201990722779464087492, −10.00897906059442975945557799449, −9.199285449098020112707744095615, −8.586035086390875577046618142101, −8.348022452245752510652932718636, −7.41411358815069438866088842761, −7.12848729492475975368876153221, −5.93494167570556792679146489156, −5.62408906820567316369247756889, −4.48979126131931170438613378661, −4.26701904090182015855711895371, −2.81433617988223982705393068140, −1.76036344919635462659354078181,
1.76036344919635462659354078181, 2.81433617988223982705393068140, 4.26701904090182015855711895371, 4.48979126131931170438613378661, 5.62408906820567316369247756889, 5.93494167570556792679146489156, 7.12848729492475975368876153221, 7.41411358815069438866088842761, 8.348022452245752510652932718636, 8.586035086390875577046618142101, 9.199285449098020112707744095615, 10.00897906059442975945557799449, 10.80976057201990722779464087492, 10.85374953070680384436830654252, 11.69428189996392567235340776374, 11.93773832768940573178041956869, 12.66531071422897734789510686624, 13.20381345458436835000960906529, 13.87107562266462274065499943418, 14.30393638177530778992496047296
