| L(s) = 1 | + 2·3-s + 3·9-s + 8·11-s + 4·17-s − 8·19-s + 25-s + 4·27-s + 16·33-s + 20·41-s − 8·43-s − 14·49-s + 8·51-s − 16·57-s + 8·59-s − 24·67-s + 20·73-s + 2·75-s + 5·81-s − 24·83-s − 12·89-s + 4·97-s + 24·99-s + 24·107-s + 4·113-s + 26·121-s + 40·123-s + 127-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 9-s + 2.41·11-s + 0.970·17-s − 1.83·19-s + 1/5·25-s + 0.769·27-s + 2.78·33-s + 3.12·41-s − 1.21·43-s − 2·49-s + 1.12·51-s − 2.11·57-s + 1.04·59-s − 2.93·67-s + 2.34·73-s + 0.230·75-s + 5/9·81-s − 2.63·83-s − 1.27·89-s + 0.406·97-s + 2.41·99-s + 2.32·107-s + 0.376·113-s + 2.36·121-s + 3.60·123-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.161759683\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.161759683\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−9.080341213166513123570518780286, −8.438902886313495638742074033009, −8.330811446905088069540074876671, −7.60125180356652570547951985370, −7.12360478295630625617848815878, −6.67875901406960725991694880905, −6.13587545605028508104380795249, −5.85130665991850657402735978605, −4.68831052899409581492283307982, −4.38871739502786245271092399731, −3.82349173075266898382654912582, −3.39274372810076015084631955767, −2.63252540175087701435373844808, −1.82346513010869405208007428430, −1.18815390030287753587945796282,
1.18815390030287753587945796282, 1.82346513010869405208007428430, 2.63252540175087701435373844808, 3.39274372810076015084631955767, 3.82349173075266898382654912582, 4.38871739502786245271092399731, 4.68831052899409581492283307982, 5.85130665991850657402735978605, 6.13587545605028508104380795249, 6.67875901406960725991694880905, 7.12360478295630625617848815878, 7.60125180356652570547951985370, 8.330811446905088069540074876671, 8.438902886313495638742074033009, 9.080341213166513123570518780286
