L(s) = 1 | − 4-s − 4·5-s + 12·11-s + 16-s − 4·19-s + 4·20-s + 11·25-s − 20·29-s + 8·31-s − 20·41-s − 12·44-s − 2·49-s − 48·55-s − 12·59-s − 12·61-s − 64-s + 4·76-s + 16·79-s − 4·80-s + 28·89-s + 16·95-s − 11·100-s − 20·101-s − 8·109-s + 20·116-s + 86·121-s − 8·124-s + ⋯ |
L(s) = 1 | − 1/2·4-s − 1.78·5-s + 3.61·11-s + 1/4·16-s − 0.917·19-s + 0.894·20-s + 11/5·25-s − 3.71·29-s + 1.43·31-s − 3.12·41-s − 1.80·44-s − 2/7·49-s − 6.47·55-s − 1.56·59-s − 1.53·61-s − 1/8·64-s + 0.458·76-s + 1.80·79-s − 0.447·80-s + 2.96·89-s + 1.64·95-s − 1.09·100-s − 1.99·101-s − 0.766·109-s + 1.85·116-s + 7.81·121-s − 0.718·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.015784202\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.015784202\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 + T^{2} \) |
good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
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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
−9.570953681673911956927210763053, −9.567518373047799809538684250061, −9.171658757631498117001698371959, −8.641452540613260864655540861877, −8.591708887486207961250456715206, −7.84799264070709687793616234040, −7.66816962489719701244400729539, −6.93002502958946787580321173830, −6.80580627847705756631003581304, −6.31767093213638303793059446864, −5.96740314632362868663684570658, −5.09112032500399629308610618367, −4.65753041938662384480582510665, −4.18387556564618950359264905326, −3.82990033815388728334514996054, −3.56866444562316507623612059497, −3.20411803066284084213875052350, −1.78897781410393362210516794694, −1.52253880076688652049512571959, −0.46842484086153357302250841885,
0.46842484086153357302250841885, 1.52253880076688652049512571959, 1.78897781410393362210516794694, 3.20411803066284084213875052350, 3.56866444562316507623612059497, 3.82990033815388728334514996054, 4.18387556564618950359264905326, 4.65753041938662384480582510665, 5.09112032500399629308610618367, 5.96740314632362868663684570658, 6.31767093213638303793059446864, 6.80580627847705756631003581304, 6.93002502958946787580321173830, 7.66816962489719701244400729539, 7.84799264070709687793616234040, 8.591708887486207961250456715206, 8.641452540613260864655540861877, 9.171658757631498117001698371959, 9.567518373047799809538684250061, 9.570953681673911956927210763053