| L(s) = 1 | + 3-s + 7-s − 2·9-s − 6·19-s + 21-s + 8·25-s − 5·27-s − 14·29-s − 5·31-s + 5·37-s + 2·47-s + 49-s − 4·53-s − 6·57-s − 59-s − 2·63-s + 8·75-s + 81-s − 2·83-s − 14·87-s − 5·93-s + 21·103-s − 20·109-s + 5·111-s + 113-s − 3·121-s + 127-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.377·7-s − 2/3·9-s − 1.37·19-s + 0.218·21-s + 8/5·25-s − 0.962·27-s − 2.59·29-s − 0.898·31-s + 0.821·37-s + 0.291·47-s + 1/7·49-s − 0.549·53-s − 0.794·57-s − 0.130·59-s − 0.251·63-s + 0.923·75-s + 1/9·81-s − 0.219·83-s − 1.50·87-s − 0.518·93-s + 2.06·103-s − 1.91·109-s + 0.474·111-s + 0.0940·113-s − 0.272·121-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 395136 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 395136 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 7 | $C_1$ | \( 1 - T \) |
| good | 5 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 9 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 31 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 68 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 29 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 88 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 117 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 + 65 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 36 T^{2} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.542535040781039612494445646059, −7.898054090850495389633143027672, −7.64588185306719147061055876738, −7.19675747119123704152837612803, −6.48249917012102815535190335274, −6.18306273735141484451776377242, −5.47349157307834204550827486455, −5.19506344531750168930858623922, −4.48489371954854980466563637664, −3.88010194101076767996684853810, −3.49358032731442054253158539592, −2.67002041621380146562851251198, −2.22775970655050359414209924387, −1.45093646265834648153230174659, 0,
1.45093646265834648153230174659, 2.22775970655050359414209924387, 2.67002041621380146562851251198, 3.49358032731442054253158539592, 3.88010194101076767996684853810, 4.48489371954854980466563637664, 5.19506344531750168930858623922, 5.47349157307834204550827486455, 6.18306273735141484451776377242, 6.48249917012102815535190335274, 7.19675747119123704152837612803, 7.64588185306719147061055876738, 7.898054090850495389633143027672, 8.542535040781039612494445646059
