L(s) = 1 | − 4·19-s + 4·31-s + 12·41-s − 49-s − 24·59-s − 20·61-s + 24·71-s − 16·79-s − 12·89-s − 12·101-s − 28·109-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
L(s) = 1 | − 0.917·19-s + 0.718·31-s + 1.87·41-s − 1/7·49-s − 3.12·59-s − 2.56·61-s + 2.84·71-s − 1.80·79-s − 1.27·89-s − 1.19·101-s − 2.68·109-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.769·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39690000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39690000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8140439887\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8140439887\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
good | 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−8.281716469057777371140307637039, −7.80586011148143379981332587606, −7.64645197101774600023572621347, −7.20046675861279683197960201975, −6.68204145814409268198327771423, −6.35704843202827958378729047479, −6.32443826020287419071147102847, −5.61639308039263914779074647007, −5.52323240729834669004304236979, −4.95553524026764991462212641778, −4.48993798539236579646064759631, −4.21232326030601155531998486483, −4.05596516642667854241614003148, −3.22078179842749272114688757198, −3.07300354767038319654858062063, −2.50806991864824026005132246761, −2.20369118026938981591620748863, −1.36199919330163688168975755910, −1.28261505809220118755366957129, −0.22978787488470587200278762617,
0.22978787488470587200278762617, 1.28261505809220118755366957129, 1.36199919330163688168975755910, 2.20369118026938981591620748863, 2.50806991864824026005132246761, 3.07300354767038319654858062063, 3.22078179842749272114688757198, 4.05596516642667854241614003148, 4.21232326030601155531998486483, 4.48993798539236579646064759631, 4.95553524026764991462212641778, 5.52323240729834669004304236979, 5.61639308039263914779074647007, 6.32443826020287419071147102847, 6.35704843202827958378729047479, 6.68204145814409268198327771423, 7.20046675861279683197960201975, 7.64645197101774600023572621347, 7.80586011148143379981332587606, 8.281716469057777371140307637039