L(s) = 1 | − 4-s − 9-s + 8·11-s + 16-s + 12·19-s + 8·29-s + 36-s − 4·41-s − 8·44-s + 10·49-s − 8·59-s − 20·61-s − 64-s − 16·71-s − 12·76-s − 16·79-s + 81-s + 28·89-s − 8·99-s − 32·101-s − 24·109-s − 8·116-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
L(s) = 1 | − 1/2·4-s − 1/3·9-s + 2.41·11-s + 1/4·16-s + 2.75·19-s + 1.48·29-s + 1/6·36-s − 0.624·41-s − 1.20·44-s + 10/7·49-s − 1.04·59-s − 2.56·61-s − 1/8·64-s − 1.89·71-s − 1.37·76-s − 1.80·79-s + 1/9·81-s + 2.96·89-s − 0.804·99-s − 3.18·101-s − 2.29·109-s − 0.742·116-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3802500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3802500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.789160324\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.789160324\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + T^{2} \) |
good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
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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
−9.266852972621859906261207318532, −9.201104856662972777511994389222, −8.773660749318981909745276051264, −8.129834527985908072697693270206, −7.978446355489964497448618663604, −7.30537283870904236891044208165, −7.06738334483959098178049173674, −6.67978329200774810527219598183, −6.16734004398887691673910083590, −5.76272696304978056234992772970, −5.50212621464203298033577285997, −4.72266480019808520811833735313, −4.60241149650481552731327675004, −3.98457387246634535008300508367, −3.60063119545053609521505864535, −2.93028893992715558285640521905, −2.91304486896806243263708883554, −1.53772617662272086988607589872, −1.40899513089758659861335663546, −0.68983782162457881447637227363,
0.68983782162457881447637227363, 1.40899513089758659861335663546, 1.53772617662272086988607589872, 2.91304486896806243263708883554, 2.93028893992715558285640521905, 3.60063119545053609521505864535, 3.98457387246634535008300508367, 4.60241149650481552731327675004, 4.72266480019808520811833735313, 5.50212621464203298033577285997, 5.76272696304978056234992772970, 6.16734004398887691673910083590, 6.67978329200774810527219598183, 7.06738334483959098178049173674, 7.30537283870904236891044208165, 7.978446355489964497448618663604, 8.129834527985908072697693270206, 8.773660749318981909745276051264, 9.201104856662972777511994389222, 9.266852972621859906261207318532