L(s) = 1 | − 2-s + 3-s + 4-s + 4·5-s − 6-s − 8-s + 9-s − 4·10-s + 2·11-s + 12-s − 2·13-s + 4·15-s + 16-s − 2·17-s − 18-s + 2·19-s + 4·20-s − 2·22-s + 23-s − 24-s + 11·25-s + 2·26-s + 27-s − 6·29-s − 4·30-s − 32-s + 2·33-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.78·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 1.26·10-s + 0.603·11-s + 0.288·12-s − 0.554·13-s + 1.03·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s + 0.458·19-s + 0.894·20-s − 0.426·22-s + 0.208·23-s − 0.204·24-s + 11/5·25-s + 0.392·26-s + 0.192·27-s − 1.11·29-s − 0.730·30-s − 0.176·32-s + 0.348·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.821592117\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.821592117\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−8.084630462795010854704711347484, −7.23804854413624616585884703178, −6.70397249135196257022116673694, −5.95135751264997398987460134229, −5.35604437544477754050638196562, −4.39958506189540498155980890034, −3.28797086113863268133699742260, −2.39679006058724391013043676749, −1.91478430921163146124419308261, −0.967516248089941926796279540286,
0.967516248089941926796279540286, 1.91478430921163146124419308261, 2.39679006058724391013043676749, 3.28797086113863268133699742260, 4.39958506189540498155980890034, 5.35604437544477754050638196562, 5.95135751264997398987460134229, 6.70397249135196257022116673694, 7.23804854413624616585884703178, 8.084630462795010854704711347484