L(s) = 1 | − 2-s − 4-s + 2·5-s − 4·7-s + 3·8-s − 3·9-s − 2·10-s − 2·13-s + 4·14-s − 16-s − 17-s + 3·18-s + 4·19-s − 2·20-s − 25-s + 2·26-s + 4·28-s + 6·29-s + 4·31-s − 5·32-s + 34-s − 8·35-s + 3·36-s + 2·37-s − 4·38-s + 6·40-s − 6·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.894·5-s − 1.51·7-s + 1.06·8-s − 9-s − 0.632·10-s − 0.554·13-s + 1.06·14-s − 1/4·16-s − 0.242·17-s + 0.707·18-s + 0.917·19-s − 0.447·20-s − 1/5·25-s + 0.392·26-s + 0.755·28-s + 1.11·29-s + 0.718·31-s − 0.883·32-s + 0.171·34-s − 1.35·35-s + 1/2·36-s + 0.328·37-s − 0.648·38-s + 0.948·40-s − 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8993 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8993 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5725261833\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5725261833\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 + T \) |
| 23 | \( 1 \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−7.912659157922053390038590689853, −7.01151153044947740224404912789, −6.42056353960850767122553958534, −5.74150827551459653725377169281, −5.11937413954888395875838280742, −4.25736803849629980048147072986, −3.17017478843378494518281039714, −2.73518417332765067838977435868, −1.56963191781081214842534969484, −0.41030662503357199003514301336,
0.41030662503357199003514301336, 1.56963191781081214842534969484, 2.73518417332765067838977435868, 3.17017478843378494518281039714, 4.25736803849629980048147072986, 5.11937413954888395875838280742, 5.74150827551459653725377169281, 6.42056353960850767122553958534, 7.01151153044947740224404912789, 7.912659157922053390038590689853