| L(s) = 1 | − 2·5-s + 6·9-s − 8·19-s − 25-s + 4·29-s + 16·31-s + 12·41-s − 12·45-s − 49-s + 8·59-s + 12·61-s − 24·71-s − 8·79-s + 27·81-s + 20·89-s + 16·95-s + 36·101-s + 28·109-s − 22·121-s + 12·125-s + 127-s + 131-s + 137-s + 139-s − 8·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 2·9-s − 1.83·19-s − 1/5·25-s + 0.742·29-s + 2.87·31-s + 1.87·41-s − 1.78·45-s − 1/7·49-s + 1.04·59-s + 1.53·61-s − 2.84·71-s − 0.900·79-s + 3·81-s + 2.11·89-s + 1.64·95-s + 3.58·101-s + 2.68·109-s − 2·121-s + 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.664·145-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5017600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5017600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.548850792\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.548850792\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 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
−8.955619885332357417309218444663, −8.947464972551267269049887564185, −8.410911655944872278933149410750, −8.037338517965774741813117061386, −7.54498922559323397897075835291, −7.51161343000160009355030276595, −6.86748832236194860180600193931, −6.60028161874464442817477133929, −6.08328048998139571672241179349, −6.00126502982397296958586899017, −4.88100801294747354385098416517, −4.79862045688158113138469808783, −4.23292128106918280694979826804, −4.18776467257530117318188829476, −3.65576844231558436063603662451, −2.99111829478864071898903528864, −2.37069313663838171317632311511, −1.97587355382782275594029392529, −1.11125498281815123418165821986, −0.65356668859271714552366025625,
0.65356668859271714552366025625, 1.11125498281815123418165821986, 1.97587355382782275594029392529, 2.37069313663838171317632311511, 2.99111829478864071898903528864, 3.65576844231558436063603662451, 4.18776467257530117318188829476, 4.23292128106918280694979826804, 4.79862045688158113138469808783, 4.88100801294747354385098416517, 6.00126502982397296958586899017, 6.08328048998139571672241179349, 6.60028161874464442817477133929, 6.86748832236194860180600193931, 7.51161343000160009355030276595, 7.54498922559323397897075835291, 8.037338517965774741813117061386, 8.410911655944872278933149410750, 8.947464972551267269049887564185, 8.955619885332357417309218444663
