L(s) = 1 | + 2-s + 3-s + 4-s − 1.56·5-s + 6-s + 1.56·7-s + 8-s + 9-s − 1.56·10-s − 4·11-s + 12-s − 4·13-s + 1.56·14-s − 1.56·15-s + 16-s − 2·17-s + 18-s + 19-s − 1.56·20-s + 1.56·21-s − 4·22-s − 0.438·23-s + 24-s − 2.56·25-s − 4·26-s + 27-s + 1.56·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.698·5-s + 0.408·6-s + 0.590·7-s + 0.353·8-s + 0.333·9-s − 0.493·10-s − 1.20·11-s + 0.288·12-s − 1.10·13-s + 0.417·14-s − 0.403·15-s + 0.250·16-s − 0.485·17-s + 0.235·18-s + 0.229·19-s − 0.349·20-s + 0.340·21-s − 0.852·22-s − 0.0914·23-s + 0.204·24-s − 0.512·25-s − 0.784·26-s + 0.192·27-s + 0.295·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 53 | \( 1 - T \) |
good | 5 | \( 1 + 1.56T + 5T^{2} \) |
| 7 | \( 1 - 1.56T + 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 23 | \( 1 + 0.438T + 23T^{2} \) |
| 29 | \( 1 - 6.68T + 29T^{2} \) |
| 31 | \( 1 + 0.438T + 31T^{2} \) |
| 37 | \( 1 - 7.12T + 37T^{2} \) |
| 41 | \( 1 + 5.12T + 41T^{2} \) |
| 43 | \( 1 + 9.56T + 43T^{2} \) |
| 47 | \( 1 - 8.24T + 47T^{2} \) |
| 59 | \( 1 + 7.80T + 59T^{2} \) |
| 61 | \( 1 + 13.1T + 61T^{2} \) |
| 67 | \( 1 + 0.684T + 67T^{2} \) |
| 71 | \( 1 + 6.24T + 71T^{2} \) |
| 73 | \( 1 + 10T + 73T^{2} \) |
| 79 | \( 1 + 13.1T + 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 + 8.43T + 89T^{2} \) |
| 97 | \( 1 + 8.24T + 97T^{2} \) |
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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
−7.70030703986597361774922482224, −7.21911007204958679622067833936, −6.29082075464620688190968146095, −5.33532247520502533417839105771, −4.67519703373438860751338587835, −4.23457931525039131657759230658, −3.08785492091803362225943046146, −2.62329525163853967643231351285, −1.62855078990082325544654644741, 0,
1.62855078990082325544654644741, 2.62329525163853967643231351285, 3.08785492091803362225943046146, 4.23457931525039131657759230658, 4.67519703373438860751338587835, 5.33532247520502533417839105771, 6.29082075464620688190968146095, 7.21911007204958679622067833936, 7.70030703986597361774922482224