L(s) = 1 | + 3-s + 4.11·7-s + 9-s − 3.21·11-s + 13-s − 6.05·17-s − 2.90·19-s + 4.11·21-s − 3.65·23-s + 27-s − 0.377·29-s − 1.40·31-s − 3.21·33-s − 10.0·37-s + 39-s + 1.33·41-s − 6.57·43-s + 8.11·47-s + 9.95·49-s − 6.05·51-s − 4.09·53-s − 2.90·57-s − 14.2·59-s − 2.57·61-s + 4.11·63-s − 7.02·67-s − 3.65·69-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1.55·7-s + 0.333·9-s − 0.969·11-s + 0.277·13-s − 1.46·17-s − 0.666·19-s + 0.898·21-s − 0.762·23-s + 0.192·27-s − 0.0701·29-s − 0.252·31-s − 0.559·33-s − 1.65·37-s + 0.160·39-s + 0.208·41-s − 1.00·43-s + 1.18·47-s + 1.42·49-s − 0.847·51-s − 0.562·53-s − 0.384·57-s − 1.85·59-s − 0.329·61-s + 0.518·63-s − 0.857·67-s − 0.440·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 - 4.11T + 7T^{2} \) |
| 11 | \( 1 + 3.21T + 11T^{2} \) |
| 17 | \( 1 + 6.05T + 17T^{2} \) |
| 19 | \( 1 + 2.90T + 19T^{2} \) |
| 23 | \( 1 + 3.65T + 23T^{2} \) |
| 29 | \( 1 + 0.377T + 29T^{2} \) |
| 31 | \( 1 + 1.40T + 31T^{2} \) |
| 37 | \( 1 + 10.0T + 37T^{2} \) |
| 41 | \( 1 - 1.33T + 41T^{2} \) |
| 43 | \( 1 + 6.57T + 43T^{2} \) |
| 47 | \( 1 - 8.11T + 47T^{2} \) |
| 53 | \( 1 + 4.09T + 53T^{2} \) |
| 59 | \( 1 + 14.2T + 59T^{2} \) |
| 61 | \( 1 + 2.57T + 61T^{2} \) |
| 67 | \( 1 + 7.02T + 67T^{2} \) |
| 71 | \( 1 - 7.19T + 71T^{2} \) |
| 73 | \( 1 + 7.46T + 73T^{2} \) |
| 79 | \( 1 - 7.13T + 79T^{2} \) |
| 83 | \( 1 + 10.9T + 83T^{2} \) |
| 89 | \( 1 + 18.5T + 89T^{2} \) |
| 97 | \( 1 - 12.2T + 97T^{2} \) |
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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
−7.62298124695175213327211594769, −7.01473243499615853878854932897, −6.10339624602389522349299584395, −5.27065723274372976964573113837, −4.60141199254077946656637275057, −4.08931592146035194875149623655, −2.99656402603008311346640098488, −2.08616424119685487402809240560, −1.62830310060678144196861963437, 0,
1.62830310060678144196861963437, 2.08616424119685487402809240560, 2.99656402603008311346640098488, 4.08931592146035194875149623655, 4.60141199254077946656637275057, 5.27065723274372976964573113837, 6.10339624602389522349299584395, 7.01473243499615853878854932897, 7.62298124695175213327211594769