| L(s) = 1 | − 3-s + 9-s − 4·11-s − 4·13-s + 12·23-s + 2·25-s − 27-s + 4·33-s − 16·37-s + 4·39-s + 4·47-s − 10·49-s + 8·59-s + 8·61-s − 12·69-s − 4·71-s − 4·73-s − 2·75-s + 81-s − 4·83-s − 4·97-s − 4·99-s + 32·107-s + 20·109-s + 16·111-s − 4·117-s − 6·121-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 1.20·11-s − 1.10·13-s + 2.50·23-s + 2/5·25-s − 0.192·27-s + 0.696·33-s − 2.63·37-s + 0.640·39-s + 0.583·47-s − 1.42·49-s + 1.04·59-s + 1.02·61-s − 1.44·69-s − 0.474·71-s − 0.468·73-s − 0.230·75-s + 1/9·81-s − 0.439·83-s − 0.406·97-s − 0.402·99-s + 3.09·107-s + 1.91·109-s + 1.51·111-s − 0.369·117-s − 0.545·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 442368 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 442368 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.025219952\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.025219952\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−8.671254765326826891497682804826, −8.172915373015799892677353765380, −7.39923236419744475792299007972, −7.25235598784162348532832243537, −6.90150753948938506235328692153, −6.32999975808964812903692189336, −5.59654427057814613520155983827, −5.16111958701561720619536259373, −4.98307741284451035740815145310, −4.49724396025832225869264707869, −3.54578243688611567456256244493, −3.07022825473137607549296482800, −2.47480380128233291684463733095, −1.67332266633815691629560142911, −0.56245071766354333363159270083,
0.56245071766354333363159270083, 1.67332266633815691629560142911, 2.47480380128233291684463733095, 3.07022825473137607549296482800, 3.54578243688611567456256244493, 4.49724396025832225869264707869, 4.98307741284451035740815145310, 5.16111958701561720619536259373, 5.59654427057814613520155983827, 6.32999975808964812903692189336, 6.90150753948938506235328692153, 7.25235598784162348532832243537, 7.39923236419744475792299007972, 8.172915373015799892677353765380, 8.671254765326826891497682804826
