| L(s) = 1 | + 4-s + 3·7-s − 8·13-s + 16-s − 2·19-s + 8·25-s + 3·28-s + 4·31-s − 8·37-s + 4·43-s − 8·52-s + 4·61-s + 64-s − 2·67-s + 10·73-s − 2·76-s − 14·79-s − 24·91-s + 10·97-s + 8·100-s + 16·103-s − 2·109-s + 3·112-s + 14·121-s + 4·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 1.13·7-s − 2.21·13-s + 1/4·16-s − 0.458·19-s + 8/5·25-s + 0.566·28-s + 0.718·31-s − 1.31·37-s + 0.609·43-s − 1.10·52-s + 0.512·61-s + 1/8·64-s − 0.244·67-s + 1.17·73-s − 0.229·76-s − 1.57·79-s − 2.51·91-s + 1.01·97-s + 4/5·100-s + 1.57·103-s − 0.191·109-s + 0.283·112-s + 1.27·121-s + 0.359·124-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900396 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900396 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.261497976\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.261497976\) |
| \(L(\frac{3}{2})\) |
|
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | | \( 1 \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 2 T + p T^{2} ) \) |
| 397 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 2 T + p T^{2} ) \) |
| good | 5 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 122 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 80 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
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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
−8.164499276060377758329309202653, −7.64797639548547643800888221306, −7.25730614406863367059061063247, −7.03021451379184098361844241487, −6.51008016961942568232308359024, −5.95817456856562742090103407625, −5.26467743805011825382482172881, −5.03035903775480568895224919667, −4.60410972052535307565081709169, −4.18868184804147197583769508376, −3.24279714313646565226875914848, −2.82692793087861267066441001783, −2.16447939942144169208117018643, −1.76278427777044204692844566966, −0.69617102321241413075817606248,
0.69617102321241413075817606248, 1.76278427777044204692844566966, 2.16447939942144169208117018643, 2.82692793087861267066441001783, 3.24279714313646565226875914848, 4.18868184804147197583769508376, 4.60410972052535307565081709169, 5.03035903775480568895224919667, 5.26467743805011825382482172881, 5.95817456856562742090103407625, 6.51008016961942568232308359024, 7.03021451379184098361844241487, 7.25730614406863367059061063247, 7.64797639548547643800888221306, 8.164499276060377758329309202653
