L(s) = 1 | + 2·2-s + 2·4-s − 5·7-s − 9-s − 6·11-s − 10·14-s − 4·16-s − 2·18-s − 12·22-s − 4·23-s − 3·25-s − 10·28-s + 2·29-s − 8·32-s − 2·36-s − 6·37-s − 12·44-s − 8·46-s + 18·49-s − 6·50-s + 2·53-s + 4·58-s + 5·63-s − 8·64-s − 67-s + 2·71-s − 12·74-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 4-s − 1.88·7-s − 1/3·9-s − 1.80·11-s − 2.67·14-s − 16-s − 0.471·18-s − 2.55·22-s − 0.834·23-s − 3/5·25-s − 1.88·28-s + 0.371·29-s − 1.41·32-s − 1/3·36-s − 0.986·37-s − 1.80·44-s − 1.17·46-s + 18/7·49-s − 0.848·50-s + 0.274·53-s + 0.525·58-s + 0.629·63-s − 64-s − 0.122·67-s + 0.237·71-s − 1.39·74-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 853776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 853776 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.013099946\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.013099946\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 7 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| 11 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
good | 5 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 11 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 35 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 71 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 21 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 107 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 37 T^{2} + p^{2} T^{4} \) |
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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
−8.148360787377756783960579641538, −7.66657287289099020522924014999, −7.05290258881509115254089049461, −6.82887139876270587854125304671, −6.17324608575687128870111201303, −5.93247645442844181763325675630, −5.48264148375570867199907669293, −5.14437926980385986395867403480, −4.47637194710682231002839543900, −3.96942376784484647534584462005, −3.41132652410486665374980615055, −3.06419486027480802927699159185, −2.56441439605749691831116722253, −2.05810142864732361546642741250, −0.34559870785058827821464289394,
0.34559870785058827821464289394, 2.05810142864732361546642741250, 2.56441439605749691831116722253, 3.06419486027480802927699159185, 3.41132652410486665374980615055, 3.96942376784484647534584462005, 4.47637194710682231002839543900, 5.14437926980385986395867403480, 5.48264148375570867199907669293, 5.93247645442844181763325675630, 6.17324608575687128870111201303, 6.82887139876270587854125304671, 7.05290258881509115254089049461, 7.66657287289099020522924014999, 8.148360787377756783960579641538