| L(s) = 1 | + 3-s − 4-s + 7-s − 2·9-s − 12-s + 7·13-s − 3·16-s − 5·19-s + 21-s − 4·25-s − 5·27-s − 28-s − 2·31-s + 2·36-s − 5·37-s + 7·39-s + 7·43-s − 3·48-s − 11·49-s − 7·52-s − 5·57-s + 7·61-s − 2·63-s + 7·64-s − 2·67-s + 4·73-s − 4·75-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1/2·4-s + 0.377·7-s − 2/3·9-s − 0.288·12-s + 1.94·13-s − 3/4·16-s − 1.14·19-s + 0.218·21-s − 4/5·25-s − 0.962·27-s − 0.188·28-s − 0.359·31-s + 1/3·36-s − 0.821·37-s + 1.12·39-s + 1.06·43-s − 0.433·48-s − 1.57·49-s − 0.970·52-s − 0.662·57-s + 0.896·61-s − 0.251·63-s + 7/8·64-s − 0.244·67-s + 0.468·73-s − 0.461·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4707 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4707 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8791767314\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8791767314\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 - T + p T^{2} \) |
| 523 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 16 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 11 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 53 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 11 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 73 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + T + p T^{2} ) \) |
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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
−12.42280676542951975434731679550, −11.55589178501301153051960816799, −11.12637773352536732897075103285, −10.74900776235841358543282745690, −9.833100974296925426993053631929, −9.114580890461426692990919620990, −8.666653697942382863852150212443, −8.309548695661207713639883239668, −7.60863552653922805566159998576, −6.52254957549357141104931271872, −6.02068222102210825933932622019, −5.10785726117687708324830895356, −4.11109856344366557420380481343, −3.47394517901535684699176941038, −2.07766409559987400500391063493,
2.07766409559987400500391063493, 3.47394517901535684699176941038, 4.11109856344366557420380481343, 5.10785726117687708324830895356, 6.02068222102210825933932622019, 6.52254957549357141104931271872, 7.60863552653922805566159998576, 8.309548695661207713639883239668, 8.666653697942382863852150212443, 9.114580890461426692990919620990, 9.833100974296925426993053631929, 10.74900776235841358543282745690, 11.12637773352536732897075103285, 11.55589178501301153051960816799, 12.42280676542951975434731679550
