| L(s) = 1 | − 1.09·3-s − 0.908·5-s + 2.29·7-s − 1.80·9-s + 4.37·13-s + 0.992·15-s − 5.06·17-s − 19-s − 2.50·21-s − 0.664·23-s − 4.17·25-s + 5.25·27-s − 3.49·29-s + 9.86·31-s − 2.08·35-s − 3.52·37-s − 4.78·39-s + 4.72·41-s + 2.17·43-s + 1.64·45-s + 3.67·47-s − 1.74·49-s + 5.53·51-s + 4.40·53-s + 1.09·57-s − 4.08·59-s − 2.18·61-s + ⋯ |
| L(s) = 1 | − 0.630·3-s − 0.406·5-s + 0.866·7-s − 0.602·9-s + 1.21·13-s + 0.256·15-s − 1.22·17-s − 0.229·19-s − 0.546·21-s − 0.138·23-s − 0.834·25-s + 1.01·27-s − 0.649·29-s + 1.77·31-s − 0.352·35-s − 0.580·37-s − 0.765·39-s + 0.738·41-s + 0.332·43-s + 0.244·45-s + 0.536·47-s − 0.248·49-s + 0.774·51-s + 0.605·53-s + 0.144·57-s − 0.531·59-s − 0.279·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 1.09T + 3T^{2} \) |
| 5 | \( 1 + 0.908T + 5T^{2} \) |
| 7 | \( 1 - 2.29T + 7T^{2} \) |
| 13 | \( 1 - 4.37T + 13T^{2} \) |
| 17 | \( 1 + 5.06T + 17T^{2} \) |
| 23 | \( 1 + 0.664T + 23T^{2} \) |
| 29 | \( 1 + 3.49T + 29T^{2} \) |
| 31 | \( 1 - 9.86T + 31T^{2} \) |
| 37 | \( 1 + 3.52T + 37T^{2} \) |
| 41 | \( 1 - 4.72T + 41T^{2} \) |
| 43 | \( 1 - 2.17T + 43T^{2} \) |
| 47 | \( 1 - 3.67T + 47T^{2} \) |
| 53 | \( 1 - 4.40T + 53T^{2} \) |
| 59 | \( 1 + 4.08T + 59T^{2} \) |
| 61 | \( 1 + 2.18T + 61T^{2} \) |
| 67 | \( 1 - 2.51T + 67T^{2} \) |
| 71 | \( 1 - 4.51T + 71T^{2} \) |
| 73 | \( 1 + 9.35T + 73T^{2} \) |
| 79 | \( 1 + 4.73T + 79T^{2} \) |
| 83 | \( 1 + 12.8T + 83T^{2} \) |
| 89 | \( 1 + 13.4T + 89T^{2} \) |
| 97 | \( 1 - 15.1T + 97T^{2} \) |
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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
−7.38332681352006591170218070852, −6.56125561834114823801131966493, −5.99363527909951332285332563088, −5.41500633931161133398212194552, −4.47880822358353290955308500603, −4.10849224812750086207195065044, −3.04584654833934275747175331022, −2.11801570228289261098433344085, −1.12378536241084558834919594658, 0,
1.12378536241084558834919594658, 2.11801570228289261098433344085, 3.04584654833934275747175331022, 4.10849224812750086207195065044, 4.47880822358353290955308500603, 5.41500633931161133398212194552, 5.99363527909951332285332563088, 6.56125561834114823801131966493, 7.38332681352006591170218070852
