L(s) = 1 | − 3-s − 7-s − 2·9-s + 3·11-s − 6·17-s + 2·19-s + 21-s − 6·23-s + 5·27-s − 6·29-s − 4·31-s − 3·33-s − 2·37-s − 2·43-s + 49-s + 6·51-s + 6·53-s − 2·57-s + 6·59-s + 2·61-s + 2·63-s − 2·67-s + 6·69-s − 3·71-s + 7·73-s − 3·77-s − 79-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.377·7-s − 2/3·9-s + 0.904·11-s − 1.45·17-s + 0.458·19-s + 0.218·21-s − 1.25·23-s + 0.962·27-s − 1.11·29-s − 0.718·31-s − 0.522·33-s − 0.328·37-s − 0.304·43-s + 1/7·49-s + 0.840·51-s + 0.824·53-s − 0.264·57-s + 0.781·59-s + 0.256·61-s + 0.251·63-s − 0.244·67-s + 0.722·69-s − 0.356·71-s + 0.819·73-s − 0.341·77-s − 0.112·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 118300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 118300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3699386465\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3699386465\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 11 T + p T^{2} \) |
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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
−13.59422421628523, −13.11633576003300, −12.52312658783620, −12.06760680662922, −11.56438873779457, −11.28580020220782, −10.79887125911104, −10.18239932129522, −9.689842478630785, −9.098649641745111, −8.788665260036313, −8.246475800802574, −7.544835137376772, −6.963575873563449, −6.553089704058396, −6.057551388312941, −5.545948867960071, −5.103686228896367, −4.253020559697396, −3.911118249033374, −3.298752496755756, −2.485995275660231, −1.972068798719951, −1.177097871113161, −0.2005084085901751,
0.2005084085901751, 1.177097871113161, 1.972068798719951, 2.485995275660231, 3.298752496755756, 3.911118249033374, 4.253020559697396, 5.103686228896367, 5.545948867960071, 6.057551388312941, 6.553089704058396, 6.963575873563449, 7.544835137376772, 8.246475800802574, 8.788665260036313, 9.098649641745111, 9.689842478630785, 10.18239932129522, 10.79887125911104, 11.28580020220782, 11.56438873779457, 12.06760680662922, 12.52312658783620, 13.11633576003300, 13.59422421628523