| L(s) = 1 | − 4-s + 4·5-s + 12·11-s + 16-s − 12·19-s − 4·20-s + 11·25-s + 4·29-s + 8·31-s + 12·41-s − 12·44-s + 14·49-s + 48·55-s + 20·59-s − 12·61-s − 64-s + 16·71-s + 12·76-s − 32·79-s + 4·80-s − 20·89-s − 48·95-s − 11·100-s + 4·101-s + 32·109-s − 4·116-s + 86·121-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 1.78·5-s + 3.61·11-s + 1/4·16-s − 2.75·19-s − 0.894·20-s + 11/5·25-s + 0.742·29-s + 1.43·31-s + 1.87·41-s − 1.80·44-s + 2·49-s + 6.47·55-s + 2.60·59-s − 1.53·61-s − 1/8·64-s + 1.89·71-s + 1.37·76-s − 3.60·79-s + 0.447·80-s − 2.11·89-s − 4.92·95-s − 1.09·100-s + 0.398·101-s + 3.06·109-s − 0.371·116-s + 7.81·121-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
| \(L(1)\) |
\(\approx\) |
\(3.929976309\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.929976309\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 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 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−9.866273678815799772379028715532, −9.565485540683676043501666521964, −9.050109140299954058908238931694, −8.935823235095889709165456676089, −8.439186645222602778280501127877, −8.424394013694957563115028387131, −7.16263961284405065907691354893, −7.03234729684659521237254278050, −6.37564738959050777857104004525, −6.29737015089268719595314095240, −6.00010299627369731866157783242, −5.49352698529555080795954766621, −4.58922386046172736197782588887, −4.29953957465948577963493537750, −4.11222484446782719110772750552, −3.44294927574075611608327444294, −2.40961131458569387104700177429, −2.26295244575655760729251193197, −1.26626032266537807946711357262, −1.08316327073093154542097857508,
1.08316327073093154542097857508, 1.26626032266537807946711357262, 2.26295244575655760729251193197, 2.40961131458569387104700177429, 3.44294927574075611608327444294, 4.11222484446782719110772750552, 4.29953957465948577963493537750, 4.58922386046172736197782588887, 5.49352698529555080795954766621, 6.00010299627369731866157783242, 6.29737015089268719595314095240, 6.37564738959050777857104004525, 7.03234729684659521237254278050, 7.16263961284405065907691354893, 8.424394013694957563115028387131, 8.439186645222602778280501127877, 8.935823235095889709165456676089, 9.050109140299954058908238931694, 9.565485540683676043501666521964, 9.866273678815799772379028715532
