| L(s) = 1 | − 2·2-s + 3·4-s + 7-s − 4·8-s − 6·9-s + 8·11-s − 2·14-s + 5·16-s + 12·18-s − 16·22-s + 3·28-s + 12·29-s − 6·32-s − 18·36-s + 20·37-s − 8·43-s + 24·44-s + 49-s + 4·53-s − 4·56-s − 24·58-s − 6·63-s + 7·64-s + 24·67-s − 32·71-s + 24·72-s − 40·74-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s + 0.377·7-s − 1.41·8-s − 2·9-s + 2.41·11-s − 0.534·14-s + 5/4·16-s + 2.82·18-s − 3.41·22-s + 0.566·28-s + 2.22·29-s − 1.06·32-s − 3·36-s + 3.28·37-s − 1.21·43-s + 3.61·44-s + 1/7·49-s + 0.549·53-s − 0.534·56-s − 3.15·58-s − 0.755·63-s + 7/8·64-s + 2.93·67-s − 3.79·71-s + 2.82·72-s − 4.64·74-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 857500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 857500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.118759399\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.118759399\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( 1 - T \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.301913068586907274040785472708, −8.089729535357517933440155073370, −7.42472757379966149042918376603, −6.82123838368350264589622397869, −6.56718920518778472538358027386, −6.01430128602422585899125610915, −5.91548647651224347146838875772, −5.14037714155681614978134696209, −4.36823223624486251008730105784, −4.03694373686491315845893969603, −2.98711940319117381089884417433, −2.95779572995708486306206498227, −2.11286590149397096544933120998, −1.27008084899723889248060652436, −0.71479745199007503910946566518,
0.71479745199007503910946566518, 1.27008084899723889248060652436, 2.11286590149397096544933120998, 2.95779572995708486306206498227, 2.98711940319117381089884417433, 4.03694373686491315845893969603, 4.36823223624486251008730105784, 5.14037714155681614978134696209, 5.91548647651224347146838875772, 6.01430128602422585899125610915, 6.56718920518778472538358027386, 6.82123838368350264589622397869, 7.42472757379966149042918376603, 8.089729535357517933440155073370, 8.301913068586907274040785472708
