| L(s) = 1 | − 9·7-s − 5·13-s + 6·19-s + 10·25-s − 12·37-s − 13·43-s + 47·49-s − 13·61-s + 21·67-s − 26·79-s + 45·91-s − 33·97-s + 26·103-s − 11·121-s + 127-s + 131-s − 54·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 12·169-s + 173-s − 90·175-s + ⋯ |
| L(s) = 1 | − 3.40·7-s − 1.38·13-s + 1.37·19-s + 2·25-s − 1.97·37-s − 1.98·43-s + 47/7·49-s − 1.66·61-s + 2.56·67-s − 2.92·79-s + 4.71·91-s − 3.35·97-s + 2.56·103-s − 121-s + 0.0887·127-s + 0.0873·131-s − 4.68·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.923·169-s + 0.0760·173-s − 6.80·175-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3504384 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3504384 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2761026919\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2761026919\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) |
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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
−9.625404835215112473600895791159, −9.191828061747497082956040237728, −8.761757896854679554068394627585, −8.415828913659778977367154055652, −7.74533690991497304844805343422, −7.14951754536012708732776646912, −6.99117279212361253848842526326, −6.73439692079206936648473774906, −6.48911821246867901283856990383, −5.80869676095609006725140305897, −5.44932253342109961711376108924, −5.09010904166559153909563342951, −4.52000787618939290002677192193, −3.85673828228338419254397576347, −3.32246481759121778041600236476, −2.98776449439593926433129434646, −2.97947230813279821631773916377, −2.14947045124176899687830411569, −1.14214071696002958645883226899, −0.20903242480536027883324706855,
0.20903242480536027883324706855, 1.14214071696002958645883226899, 2.14947045124176899687830411569, 2.97947230813279821631773916377, 2.98776449439593926433129434646, 3.32246481759121778041600236476, 3.85673828228338419254397576347, 4.52000787618939290002677192193, 5.09010904166559153909563342951, 5.44932253342109961711376108924, 5.80869676095609006725140305897, 6.48911821246867901283856990383, 6.73439692079206936648473774906, 6.99117279212361253848842526326, 7.14951754536012708732776646912, 7.74533690991497304844805343422, 8.415828913659778977367154055652, 8.761757896854679554068394627585, 9.191828061747497082956040237728, 9.625404835215112473600895791159
