| L(s) = 1 | − 2-s + 6·3-s + 2·4-s + 3·5-s − 6·6-s − 5·8-s + 21·9-s − 3·10-s − 6·11-s + 12·12-s + 2·13-s + 18·15-s + 5·16-s − 2·17-s − 21·18-s + 2·19-s + 6·20-s + 6·22-s − 30·24-s + 5·25-s − 2·26-s + 54·27-s − 7·29-s − 18·30-s + 3·31-s − 10·32-s − 36·33-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 3.46·3-s + 4-s + 1.34·5-s − 2.44·6-s − 1.76·8-s + 7·9-s − 0.948·10-s − 1.80·11-s + 3.46·12-s + 0.554·13-s + 4.64·15-s + 5/4·16-s − 0.485·17-s − 4.94·18-s + 0.458·19-s + 1.34·20-s + 1.27·22-s − 6.12·24-s + 25-s − 0.392·26-s + 10.3·27-s − 1.29·29-s − 3.28·30-s + 0.538·31-s − 1.76·32-s − 6.26·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405769 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405769 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.042431745\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.042431745\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | | \( 1 \) |
| 13 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 - p T + p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 7 T + 20 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 3 T - 22 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 7 T + 6 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - T - 46 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 3 T - 44 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 13 T + 98 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 13 T + 96 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 3 T - 70 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) |
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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
−10.42149659983095379738553630387, −10.08993635598612210373246806177, −9.778685184786831876448337038634, −9.225572483665266828648821705129, −8.989526376112955629874363703719, −8.900877084746194396536859604799, −8.139316254233997057346155697860, −8.007832841253923214119331855791, −7.56306674795424406332456218611, −7.00336509204686665321898464106, −6.72426177460275896711449148052, −5.71406097845946011556786003564, −5.65715707973529754724028725698, −4.60003071303646567570301607707, −3.80813654828250170717958005537, −3.28411039031949028525051511701, −2.79304304217424092196289102658, −2.43417156954458916176106786594, −2.16984235531105985857885849370, −1.45000882721024066151316439259,
1.45000882721024066151316439259, 2.16984235531105985857885849370, 2.43417156954458916176106786594, 2.79304304217424092196289102658, 3.28411039031949028525051511701, 3.80813654828250170717958005537, 4.60003071303646567570301607707, 5.65715707973529754724028725698, 5.71406097845946011556786003564, 6.72426177460275896711449148052, 7.00336509204686665321898464106, 7.56306674795424406332456218611, 8.007832841253923214119331855791, 8.139316254233997057346155697860, 8.900877084746194396536859604799, 8.989526376112955629874363703719, 9.225572483665266828648821705129, 9.778685184786831876448337038634, 10.08993635598612210373246806177, 10.42149659983095379738553630387
