| L(s) = 1 | − 2-s + 3-s + 2·5-s − 6-s + 3·7-s + 8-s − 2·10-s − 11-s + 5·13-s − 3·14-s + 2·15-s − 16-s + 5·19-s + 3·21-s + 22-s + 4·23-s + 24-s + 3·25-s − 5·26-s − 27-s − 2·30-s + 20·31-s − 33-s + 6·35-s + 37-s − 5·38-s + 5·39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.894·5-s − 0.408·6-s + 1.13·7-s + 0.353·8-s − 0.632·10-s − 0.301·11-s + 1.38·13-s − 0.801·14-s + 0.516·15-s − 1/4·16-s + 1.14·19-s + 0.654·21-s + 0.213·22-s + 0.834·23-s + 0.204·24-s + 3/5·25-s − 0.980·26-s − 0.192·27-s − 0.365·30-s + 3.59·31-s − 0.174·33-s + 1.01·35-s + 0.164·37-s − 0.811·38-s + 0.800·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.010491062\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.010491062\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 3 T + 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + T - 10 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 2 T - 39 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 12 T + 77 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 2 T - 67 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + T - 88 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 12 T + 47 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
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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
−11.40655522443016623374508753724, −11.12404713016780228056349221395, −10.34764191178301719309077505078, −10.27043887772521701990008525581, −9.515933477543360842369224289040, −9.401552158342165510545022240980, −8.687890017258664319130095368439, −8.322000406059020873032228095327, −7.993971716072043059200473055327, −7.73415438426780140686718026043, −6.62330776080705882361830139222, −6.58710689688921022555872509310, −5.83154900473575940787590018557, −5.07785596282092560504841060970, −4.83044610181464682938803592506, −4.13057929401455010479128368689, −2.94655593616989399592448213886, −2.94043778047944064439256348064, −1.50504020283557303662799104360, −1.33862244464039955174558479217,
1.33862244464039955174558479217, 1.50504020283557303662799104360, 2.94043778047944064439256348064, 2.94655593616989399592448213886, 4.13057929401455010479128368689, 4.83044610181464682938803592506, 5.07785596282092560504841060970, 5.83154900473575940787590018557, 6.58710689688921022555872509310, 6.62330776080705882361830139222, 7.73415438426780140686718026043, 7.993971716072043059200473055327, 8.322000406059020873032228095327, 8.687890017258664319130095368439, 9.401552158342165510545022240980, 9.515933477543360842369224289040, 10.27043887772521701990008525581, 10.34764191178301719309077505078, 11.12404713016780228056349221395, 11.40655522443016623374508753724
