| L(s) = 1 | − 2·3-s − 3·4-s + 3·9-s + 6·12-s − 2·13-s + 5·16-s + 4·17-s + 25-s − 4·27-s − 4·29-s − 9·36-s + 4·39-s + 8·43-s − 10·48-s − 14·49-s − 8·51-s + 6·52-s − 20·53-s − 4·61-s − 3·64-s − 12·68-s − 2·75-s + 5·81-s + 8·87-s − 3·100-s + 12·101-s − 32·103-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 3/2·4-s + 9-s + 1.73·12-s − 0.554·13-s + 5/4·16-s + 0.970·17-s + 1/5·25-s − 0.769·27-s − 0.742·29-s − 3/2·36-s + 0.640·39-s + 1.21·43-s − 1.44·48-s − 2·49-s − 1.12·51-s + 0.832·52-s − 2.74·53-s − 0.512·61-s − 3/8·64-s − 1.45·68-s − 0.230·75-s + 5/9·81-s + 0.857·87-s − 0.299·100-s + 1.19·101-s − 3.15·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 13 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 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
−9.910992058599380201232598700641, −9.523451675812284265792380668213, −9.333388465635554502086321327335, −8.526905706632633062975018882770, −7.76910750368042878757814163764, −7.66488013441745380243230842523, −6.63078439380576833175908493602, −6.20481649249773216903893847569, −5.27152076797277019902163228084, −5.23920392624592057055772361749, −4.46342990915686607443282092868, −3.92265287197659703286168465477, −3.01746980794826050429743741143, −1.41836019676092930181617969749, 0,
1.41836019676092930181617969749, 3.01746980794826050429743741143, 3.92265287197659703286168465477, 4.46342990915686607443282092868, 5.23920392624592057055772361749, 5.27152076797277019902163228084, 6.20481649249773216903893847569, 6.63078439380576833175908493602, 7.66488013441745380243230842523, 7.76910750368042878757814163764, 8.526905706632633062975018882770, 9.333388465635554502086321327335, 9.523451675812284265792380668213, 9.910992058599380201232598700641
