L(s) = 1 | − 3-s + 5-s + 1.82·7-s + 9-s − 1.41·11-s + 0.585·13-s − 15-s − 2.41·17-s − 7.41·19-s − 1.82·21-s − 23-s + 25-s − 27-s − 3.58·29-s − 31-s + 1.41·33-s + 1.82·35-s − 7.48·37-s − 0.585·39-s + 7.24·41-s − 7.65·43-s + 45-s − 6.24·47-s − 3.65·49-s + 2.41·51-s + 5.24·53-s − 1.41·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.691·7-s + 0.333·9-s − 0.426·11-s + 0.162·13-s − 0.258·15-s − 0.585·17-s − 1.70·19-s − 0.398·21-s − 0.208·23-s + 0.200·25-s − 0.192·27-s − 0.665·29-s − 0.179·31-s + 0.246·33-s + 0.309·35-s − 1.23·37-s − 0.0938·39-s + 1.13·41-s − 1.16·43-s + 0.149·45-s − 0.910·47-s − 0.522·49-s + 0.338·51-s + 0.720·53-s − 0.190·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 7 | \( 1 - 1.82T + 7T^{2} \) |
| 11 | \( 1 + 1.41T + 11T^{2} \) |
| 13 | \( 1 - 0.585T + 13T^{2} \) |
| 17 | \( 1 + 2.41T + 17T^{2} \) |
| 19 | \( 1 + 7.41T + 19T^{2} \) |
| 29 | \( 1 + 3.58T + 29T^{2} \) |
| 31 | \( 1 + T + 31T^{2} \) |
| 37 | \( 1 + 7.48T + 37T^{2} \) |
| 41 | \( 1 - 7.24T + 41T^{2} \) |
| 43 | \( 1 + 7.65T + 43T^{2} \) |
| 47 | \( 1 + 6.24T + 47T^{2} \) |
| 53 | \( 1 - 5.24T + 53T^{2} \) |
| 59 | \( 1 - 13.7T + 59T^{2} \) |
| 61 | \( 1 - 1.41T + 61T^{2} \) |
| 67 | \( 1 + 15.4T + 67T^{2} \) |
| 71 | \( 1 + 14.0T + 71T^{2} \) |
| 73 | \( 1 - 2.24T + 73T^{2} \) |
| 79 | \( 1 - 16.9T + 79T^{2} \) |
| 83 | \( 1 - 2.75T + 83T^{2} \) |
| 89 | \( 1 + 9.17T + 89T^{2} \) |
| 97 | \( 1 - 18.1T + 97T^{2} \) |
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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
−8.500823890187794046726767853016, −7.68177115475269766379648472230, −6.77567400941877199448966035723, −6.14712730096113182993513776755, −5.30381469439177252294351165271, −4.64256082963014872091510660186, −3.76639233830346217272872851444, −2.37813619167094289574387965841, −1.60217839941765170196136999364, 0,
1.60217839941765170196136999364, 2.37813619167094289574387965841, 3.76639233830346217272872851444, 4.64256082963014872091510660186, 5.30381469439177252294351165271, 6.14712730096113182993513776755, 6.77567400941877199448966035723, 7.68177115475269766379648472230, 8.500823890187794046726767853016