L(s) = 1 | + 2-s − 1.82·3-s + 4-s − 5-s − 1.82·6-s + 7-s + 8-s + 0.328·9-s − 10-s − 1.82·12-s − 2.21·13-s + 14-s + 1.82·15-s + 16-s − 7.96·17-s + 0.328·18-s − 0.915·19-s − 20-s − 1.82·21-s + 6.07·23-s − 1.82·24-s + 25-s − 2.21·26-s + 4.87·27-s + 28-s + 5.06·29-s + 1.82·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.05·3-s + 0.5·4-s − 0.447·5-s − 0.744·6-s + 0.377·7-s + 0.353·8-s + 0.109·9-s − 0.316·10-s − 0.526·12-s − 0.614·13-s + 0.267·14-s + 0.471·15-s + 0.250·16-s − 1.93·17-s + 0.0773·18-s − 0.210·19-s − 0.223·20-s − 0.398·21-s + 1.26·23-s − 0.372·24-s + 0.200·25-s − 0.434·26-s + 0.938·27-s + 0.188·28-s + 0.940·29-s + 0.333·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 3 | \( 1 + 1.82T + 3T^{2} \) |
| 13 | \( 1 + 2.21T + 13T^{2} \) |
| 17 | \( 1 + 7.96T + 17T^{2} \) |
| 19 | \( 1 + 0.915T + 19T^{2} \) |
| 23 | \( 1 - 6.07T + 23T^{2} \) |
| 29 | \( 1 - 5.06T + 29T^{2} \) |
| 31 | \( 1 - 9.33T + 31T^{2} \) |
| 37 | \( 1 - 4.32T + 37T^{2} \) |
| 41 | \( 1 + 8.99T + 41T^{2} \) |
| 43 | \( 1 - 2.76T + 43T^{2} \) |
| 47 | \( 1 - 4.22T + 47T^{2} \) |
| 53 | \( 1 + 9.33T + 53T^{2} \) |
| 59 | \( 1 - 7.19T + 59T^{2} \) |
| 61 | \( 1 + 12.6T + 61T^{2} \) |
| 67 | \( 1 + 1.29T + 67T^{2} \) |
| 71 | \( 1 - 0.0202T + 71T^{2} \) |
| 73 | \( 1 - 11.8T + 73T^{2} \) |
| 79 | \( 1 + 9.39T + 79T^{2} \) |
| 83 | \( 1 - 0.747T + 83T^{2} \) |
| 89 | \( 1 + 14.2T + 89T^{2} \) |
| 97 | \( 1 + 7.27T + 97T^{2} \) |
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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
−7.03441453839236369406788921620, −6.70856425075925496535603765500, −6.07844326298878534312235074087, −5.18966279768516864879567378695, −4.62900168774724117475754131708, −4.33124130547277825933337546245, −3.02602786217034420862124818489, −2.42557439155520253820831286658, −1.14357868511710481442022382815, 0,
1.14357868511710481442022382815, 2.42557439155520253820831286658, 3.02602786217034420862124818489, 4.33124130547277825933337546245, 4.62900168774724117475754131708, 5.18966279768516864879567378695, 6.07844326298878534312235074087, 6.70856425075925496535603765500, 7.03441453839236369406788921620