L(s) = 1 | + 2-s − 3-s + 4-s − 2.80·5-s − 6-s − 1.72·7-s + 8-s + 9-s − 2.80·10-s − 0.880·11-s − 12-s + 13-s − 1.72·14-s + 2.80·15-s + 16-s + 4.56·17-s + 18-s − 1.60·19-s − 2.80·20-s + 1.72·21-s − 0.880·22-s − 7.79·23-s − 24-s + 2.88·25-s + 26-s − 27-s − 1.72·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.25·5-s − 0.408·6-s − 0.652·7-s + 0.353·8-s + 0.333·9-s − 0.888·10-s − 0.265·11-s − 0.288·12-s + 0.277·13-s − 0.461·14-s + 0.725·15-s + 0.250·16-s + 1.10·17-s + 0.235·18-s − 0.368·19-s − 0.627·20-s + 0.376·21-s − 0.187·22-s − 1.62·23-s − 0.204·24-s + 0.577·25-s + 0.196·26-s − 0.192·27-s − 0.326·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 + T \) |
good | 5 | \( 1 + 2.80T + 5T^{2} \) |
| 7 | \( 1 + 1.72T + 7T^{2} \) |
| 11 | \( 1 + 0.880T + 11T^{2} \) |
| 17 | \( 1 - 4.56T + 17T^{2} \) |
| 19 | \( 1 + 1.60T + 19T^{2} \) |
| 23 | \( 1 + 7.79T + 23T^{2} \) |
| 29 | \( 1 - 9.56T + 29T^{2} \) |
| 31 | \( 1 - 3.05T + 31T^{2} \) |
| 37 | \( 1 - 4.09T + 37T^{2} \) |
| 41 | \( 1 + 8.26T + 41T^{2} \) |
| 43 | \( 1 - 1.93T + 43T^{2} \) |
| 47 | \( 1 - 8.49T + 47T^{2} \) |
| 53 | \( 1 + 4.93T + 53T^{2} \) |
| 59 | \( 1 - 9.18T + 59T^{2} \) |
| 61 | \( 1 - 0.950T + 61T^{2} \) |
| 67 | \( 1 - 3.50T + 67T^{2} \) |
| 71 | \( 1 - 6.29T + 71T^{2} \) |
| 73 | \( 1 + 4.17T + 73T^{2} \) |
| 79 | \( 1 + 10.9T + 79T^{2} \) |
| 83 | \( 1 + 5.52T + 83T^{2} \) |
| 89 | \( 1 - 3.08T + 89T^{2} \) |
| 97 | \( 1 - 2.54T + 97T^{2} \) |
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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
−7.41072952295640316322344676637, −6.63390406209614761585473729615, −6.12171148468369172404447190387, −5.38731908737566750947186997796, −4.54022865698187221820764416236, −3.96793469129273758944505328551, −3.34458142798970251110341024522, −2.48643565088574851748182589865, −1.11446913555108819694796087138, 0,
1.11446913555108819694796087138, 2.48643565088574851748182589865, 3.34458142798970251110341024522, 3.96793469129273758944505328551, 4.54022865698187221820764416236, 5.38731908737566750947186997796, 6.12171148468369172404447190387, 6.63390406209614761585473729615, 7.41072952295640316322344676637