| L(s) = 1 | + 3·4-s − 4·7-s − 10·13-s + 5·16-s − 3·25-s − 12·28-s + 8·31-s − 6·37-s − 12·43-s − 2·49-s − 30·52-s + 12·61-s + 3·64-s − 20·67-s + 20·73-s + 16·79-s + 40·91-s + 10·97-s − 9·100-s + 28·103-s − 10·109-s − 20·112-s + 24·124-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 3/2·4-s − 1.51·7-s − 2.77·13-s + 5/4·16-s − 3/5·25-s − 2.26·28-s + 1.43·31-s − 0.986·37-s − 1.82·43-s − 2/7·49-s − 4.16·52-s + 1.53·61-s + 3/8·64-s − 2.44·67-s + 2.34·73-s + 1.80·79-s + 4.19·91-s + 1.01·97-s − 0.899·100-s + 2.75·103-s − 0.957·109-s − 1.88·112-s + 2.15·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.338929394\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.338929394\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 27 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 75 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 43 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 110 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 115 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
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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
−10.01926955320508892018883933110, −9.914965703692168903433585252577, −9.453377280534218477108632006757, −8.936217250586578167103295900565, −8.326994454780663972586007875392, −7.80500451405150098022293393114, −7.53088307155573362187777573676, −6.97800876946857858783845001871, −6.84591754041248826153327675164, −6.32708095088322834043551689914, −6.13775654052852944620665145654, −5.33759634431304167642847462094, −4.95387901592604198404956014156, −4.54409866550446167106172668134, −3.59897321520177948699551500908, −3.21424160862695807850449637521, −2.78790442035988167127374852458, −2.21395156865506405044785996269, −1.85912584640685147780352936602, −0.46411730221702557237105906800,
0.46411730221702557237105906800, 1.85912584640685147780352936602, 2.21395156865506405044785996269, 2.78790442035988167127374852458, 3.21424160862695807850449637521, 3.59897321520177948699551500908, 4.54409866550446167106172668134, 4.95387901592604198404956014156, 5.33759634431304167642847462094, 6.13775654052852944620665145654, 6.32708095088322834043551689914, 6.84591754041248826153327675164, 6.97800876946857858783845001871, 7.53088307155573362187777573676, 7.80500451405150098022293393114, 8.326994454780663972586007875392, 8.936217250586578167103295900565, 9.453377280534218477108632006757, 9.914965703692168903433585252577, 10.01926955320508892018883933110
