L(s) = 1 | + 3-s − 2·4-s + 9-s + 11-s − 2·12-s + 2·13-s + 4·16-s + 4·19-s − 6·23-s + 27-s − 5·31-s + 33-s − 2·36-s − 11·37-s + 2·39-s − 6·41-s + 43-s − 2·44-s + 6·47-s + 4·48-s − 4·52-s − 12·53-s + 4·57-s + 6·59-s − 5·61-s − 8·64-s + 4·67-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 4-s + 1/3·9-s + 0.301·11-s − 0.577·12-s + 0.554·13-s + 16-s + 0.917·19-s − 1.25·23-s + 0.192·27-s − 0.898·31-s + 0.174·33-s − 1/3·36-s − 1.80·37-s + 0.320·39-s − 0.937·41-s + 0.152·43-s − 0.301·44-s + 0.875·47-s + 0.577·48-s − 0.554·52-s − 1.64·53-s + 0.529·57-s + 0.781·59-s − 0.640·61-s − 64-s + 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.879947126\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.879947126\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 2 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.52104613189965, −14.18343602867605, −13.76607893441307, −13.46863529826235, −12.62446197189915, −12.39824434431761, −11.73559768690527, −11.07320261917922, −10.39697387330446, −9.949234574614128, −9.371023835337035, −9.004863243657706, −8.351016491350469, −8.033197791921007, −7.330090811297471, −6.742863947697453, −5.944256557881730, −5.412252953526347, −4.822072121262208, −4.094092812794484, −3.555220307025097, −3.199270827107699, −2.049448032655484, −1.462679190539751, −0.5018492964541746,
0.5018492964541746, 1.462679190539751, 2.049448032655484, 3.199270827107699, 3.555220307025097, 4.094092812794484, 4.822072121262208, 5.412252953526347, 5.944256557881730, 6.742863947697453, 7.330090811297471, 8.033197791921007, 8.351016491350469, 9.004863243657706, 9.371023835337035, 9.949234574614128, 10.39697387330446, 11.07320261917922, 11.73559768690527, 12.39824434431761, 12.62446197189915, 13.46863529826235, 13.76607893441307, 14.18343602867605, 14.52104613189965