| L(s) = 1 | + 2-s + 3-s + 4·5-s + 6-s + 7-s − 8-s + 4·10-s + 11-s + 5·13-s + 14-s + 4·15-s − 16-s + 17-s − 19-s + 21-s + 22-s + 6·23-s − 24-s + 2·25-s + 5·26-s − 27-s − 9·29-s + 4·30-s + 4·31-s + 33-s + 34-s + 4·35-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1.78·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1.26·10-s + 0.301·11-s + 1.38·13-s + 0.267·14-s + 1.03·15-s − 1/4·16-s + 0.242·17-s − 0.229·19-s + 0.218·21-s + 0.213·22-s + 1.25·23-s − 0.204·24-s + 2/5·25-s + 0.980·26-s − 0.192·27-s − 1.67·29-s + 0.730·30-s + 0.718·31-s + 0.174·33-s + 0.171·34-s + 0.676·35-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
| \(L(1)\) |
\(\approx\) |
\(4.719916119\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.719916119\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | $C_2$ | \( 1 - T + T^{2} \) |
| 13 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - T - 10 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - T - 16 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 9 T + 52 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 5 T - 16 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 2 T - 39 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 10 T + 41 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 11 T + 60 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 8 T - 7 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 11 T + 32 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 19 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
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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
−11.01634578570366453947207962250, −10.71207294620912451786141981298, −9.828594102333713680469629293496, −9.753253675890985675267943564419, −9.373104280125711054227751266293, −8.884124869224620897261256184805, −8.347068081558194163117139992592, −8.127122846070410330602621664607, −7.37637192198163924055116237553, −6.72438510992968681155549614488, −6.33477531013229680853472617202, −5.90409527694561589222488243218, −5.50439121103564016485082952494, −5.01833616881569129907023582109, −4.45200911115750619629331998244, −3.52371657152434131793503906570, −3.47553916883469902126890237556, −2.47303353765238207728906545702, −1.90369392014556325526361997921, −1.28887603936948580206798661718,
1.28887603936948580206798661718, 1.90369392014556325526361997921, 2.47303353765238207728906545702, 3.47553916883469902126890237556, 3.52371657152434131793503906570, 4.45200911115750619629331998244, 5.01833616881569129907023582109, 5.50439121103564016485082952494, 5.90409527694561589222488243218, 6.33477531013229680853472617202, 6.72438510992968681155549614488, 7.37637192198163924055116237553, 8.127122846070410330602621664607, 8.347068081558194163117139992592, 8.884124869224620897261256184805, 9.373104280125711054227751266293, 9.753253675890985675267943564419, 9.828594102333713680469629293496, 10.71207294620912451786141981298, 11.01634578570366453947207962250
