| L(s) = 1 | − 2-s + 3·3-s − 3·6-s + 5·7-s + 8-s + 6·9-s + 3·11-s + 7·13-s − 5·14-s − 16-s + 3·17-s − 6·18-s − 7·19-s + 15·21-s − 3·22-s − 12·23-s + 3·24-s − 2·25-s − 7·26-s + 9·27-s + 3·29-s − 8·31-s + 9·33-s − 3·34-s − 12·37-s + 7·38-s + 21·39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.73·3-s − 1.22·6-s + 1.88·7-s + 0.353·8-s + 2·9-s + 0.904·11-s + 1.94·13-s − 1.33·14-s − 1/4·16-s + 0.727·17-s − 1.41·18-s − 1.60·19-s + 3.27·21-s − 0.639·22-s − 2.50·23-s + 0.612·24-s − 2/5·25-s − 1.37·26-s + 1.73·27-s + 0.557·29-s − 1.43·31-s + 1.56·33-s − 0.514·34-s − 1.97·37-s + 1.13·38-s + 3.36·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.193935521\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.193935521\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 7 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 - 7 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 12 T + 71 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 3 T + 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 9 T + 68 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 19 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 31 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 18 T + 167 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 9 T + 88 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 6 T - 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 15 T + 164 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 2 T - 93 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−10.76404171652445162436368700314, −10.35755586450647557013418473890, −10.34770923065753354751382546040, −9.425874682483268862394938997865, −9.035291318085816790716224679813, −8.739937618297979873078353338759, −8.445119464798784168124227129641, −8.101814200092245320152805719619, −7.66825729244884065434916349889, −7.39024815444517330674212964731, −6.63660508751633759085438156484, −5.81676974811283150742554884998, −5.78103953968848611458012491794, −4.42096559079614079033351548551, −4.33438896394422983349627880181, −3.83325223241200571344385514153, −3.29562297744852499127160483122, −2.05434470899961253077337944268, −1.87452374099748778201388988223, −1.27799469535505028144983064005,
1.27799469535505028144983064005, 1.87452374099748778201388988223, 2.05434470899961253077337944268, 3.29562297744852499127160483122, 3.83325223241200571344385514153, 4.33438896394422983349627880181, 4.42096559079614079033351548551, 5.78103953968848611458012491794, 5.81676974811283150742554884998, 6.63660508751633759085438156484, 7.39024815444517330674212964731, 7.66825729244884065434916349889, 8.101814200092245320152805719619, 8.445119464798784168124227129641, 8.739937618297979873078353338759, 9.035291318085816790716224679813, 9.425874682483268862394938997865, 10.34770923065753354751382546040, 10.35755586450647557013418473890, 10.76404171652445162436368700314
