L(s) = 1 | − 2-s + 4-s − 2·5-s − 8-s + 2·10-s + 4·11-s − 2·13-s + 16-s − 2·17-s − 2·20-s − 4·22-s − 25-s + 2·26-s − 8·29-s − 8·31-s − 32-s + 2·34-s + 4·37-s + 2·40-s − 10·41-s − 2·43-s + 4·44-s + 50-s − 2·52-s + 8·53-s − 8·55-s + 8·58-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.894·5-s − 0.353·8-s + 0.632·10-s + 1.20·11-s − 0.554·13-s + 1/4·16-s − 0.485·17-s − 0.447·20-s − 0.852·22-s − 1/5·25-s + 0.392·26-s − 1.48·29-s − 1.43·31-s − 0.176·32-s + 0.342·34-s + 0.657·37-s + 0.316·40-s − 1.56·41-s − 0.304·43-s + 0.603·44-s + 0.141·50-s − 0.277·52-s + 1.09·53-s − 1.07·55-s + 1.05·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69678 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69678 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 79 | \( 1 + T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−14.69674385773044, −13.95049559845017, −13.31553040510364, −12.83154477081239, −12.19305901420533, −11.76938127627145, −11.34515763877006, −11.06807674772346, −10.25933523328283, −9.768333343241987, −9.310772687362020, −8.682771727325343, −8.456727010141663, −7.551529336368969, −7.338322007211144, −6.820749181195385, −6.204331863445010, −5.522602474901016, −4.928042575422276, −3.961838652717013, −3.862293681645247, −3.112479264406057, −2.130389252622517, −1.729509488950422, −0.7279197459888474, 0,
0.7279197459888474, 1.729509488950422, 2.130389252622517, 3.112479264406057, 3.862293681645247, 3.961838652717013, 4.928042575422276, 5.522602474901016, 6.204331863445010, 6.820749181195385, 7.338322007211144, 7.551529336368969, 8.456727010141663, 8.682771727325343, 9.310772687362020, 9.768333343241987, 10.25933523328283, 11.06807674772346, 11.34515763877006, 11.76938127627145, 12.19305901420533, 12.83154477081239, 13.31553040510364, 13.95049559845017, 14.69674385773044