| L(s) = 1 | + 2·3-s + 3·4-s + 2·7-s + 3·9-s + 6·11-s + 6·12-s + 6·13-s + 5·16-s + 8·17-s − 2·19-s + 4·21-s − 8·23-s + 4·27-s + 6·28-s + 2·29-s + 12·31-s + 12·33-s + 9·36-s − 2·37-s + 12·39-s − 14·41-s − 6·43-s + 18·44-s − 8·47-s + 10·48-s − 4·49-s + 16·51-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 3/2·4-s + 0.755·7-s + 9-s + 1.80·11-s + 1.73·12-s + 1.66·13-s + 5/4·16-s + 1.94·17-s − 0.458·19-s + 0.872·21-s − 1.66·23-s + 0.769·27-s + 1.13·28-s + 0.371·29-s + 2.15·31-s + 2.08·33-s + 3/2·36-s − 0.328·37-s + 1.92·39-s − 2.18·41-s − 0.914·43-s + 2.71·44-s − 1.16·47-s + 1.44·48-s − 4/7·49-s + 2.24·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2030625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2030625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.223565340\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.223565340\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 2 T + 8 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 6 T + 24 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 + 8 T + 34 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 2 T - 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 2 T + 68 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 14 T + 124 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 6 T + 88 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 8 T + 82 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 12 T + 114 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 12 T + 126 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 12 T + 130 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 38 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 4 T + 118 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 62 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 + 18 T + 196 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 10 T + 156 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
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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
−10.00230922795690139354083738255, −9.324897704878158269419186063575, −8.643139240401713534112512363744, −8.460777693018554677663563743162, −8.272003315255411304844981485799, −7.76167606018827966499497845131, −7.49076444965092625297836881843, −6.64773428201087790529392224422, −6.64390645269505056568588734419, −6.26988906281124121992944018435, −5.81299368671187477015046218556, −5.22398748741409911909706967055, −4.35989592853501084316322633033, −4.21442661820592900103616670130, −3.50505424964921679592495139236, −3.13866586407307833967892557544, −2.87575319184643748254046897491, −1.73383429590091245764133311669, −1.54330696656937962354340596612, −1.36556345711745183265884731997,
1.36556345711745183265884731997, 1.54330696656937962354340596612, 1.73383429590091245764133311669, 2.87575319184643748254046897491, 3.13866586407307833967892557544, 3.50505424964921679592495139236, 4.21442661820592900103616670130, 4.35989592853501084316322633033, 5.22398748741409911909706967055, 5.81299368671187477015046218556, 6.26988906281124121992944018435, 6.64390645269505056568588734419, 6.64773428201087790529392224422, 7.49076444965092625297836881843, 7.76167606018827966499497845131, 8.272003315255411304844981485799, 8.460777693018554677663563743162, 8.643139240401713534112512363744, 9.324897704878158269419186063575, 10.00230922795690139354083738255
