L(s) = 1 | + 2.60·3-s − 1.79·5-s − 2.96·7-s + 3.80·9-s − 0.556·11-s + 3.76·13-s − 4.69·15-s + 1.41·17-s + 1.17·19-s − 7.72·21-s + 0.626·23-s − 1.76·25-s + 2.11·27-s − 0.654·29-s + 1.67·31-s − 1.45·33-s + 5.32·35-s − 2.59·37-s + 9.83·39-s + 6.68·41-s + 11.7·43-s − 6.85·45-s + 13.0·47-s + 1.76·49-s + 3.69·51-s + 9.66·53-s + 1.00·55-s + ⋯ |
L(s) = 1 | + 1.50·3-s − 0.804·5-s − 1.11·7-s + 1.26·9-s − 0.167·11-s + 1.04·13-s − 1.21·15-s + 0.343·17-s + 0.270·19-s − 1.68·21-s + 0.130·23-s − 0.352·25-s + 0.406·27-s − 0.121·29-s + 0.300·31-s − 0.252·33-s + 0.900·35-s − 0.426·37-s + 1.57·39-s + 1.04·41-s + 1.79·43-s − 1.02·45-s + 1.89·47-s + 0.252·49-s + 0.517·51-s + 1.32·53-s + 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.570084994\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.570084994\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 503 | \( 1 - T \) |
good | 3 | \( 1 - 2.60T + 3T^{2} \) |
| 5 | \( 1 + 1.79T + 5T^{2} \) |
| 7 | \( 1 + 2.96T + 7T^{2} \) |
| 11 | \( 1 + 0.556T + 11T^{2} \) |
| 13 | \( 1 - 3.76T + 13T^{2} \) |
| 17 | \( 1 - 1.41T + 17T^{2} \) |
| 19 | \( 1 - 1.17T + 19T^{2} \) |
| 23 | \( 1 - 0.626T + 23T^{2} \) |
| 29 | \( 1 + 0.654T + 29T^{2} \) |
| 31 | \( 1 - 1.67T + 31T^{2} \) |
| 37 | \( 1 + 2.59T + 37T^{2} \) |
| 41 | \( 1 - 6.68T + 41T^{2} \) |
| 43 | \( 1 - 11.7T + 43T^{2} \) |
| 47 | \( 1 - 13.0T + 47T^{2} \) |
| 53 | \( 1 - 9.66T + 53T^{2} \) |
| 59 | \( 1 - 5.84T + 59T^{2} \) |
| 61 | \( 1 + 6.37T + 61T^{2} \) |
| 67 | \( 1 - 9.80T + 67T^{2} \) |
| 71 | \( 1 + 14.7T + 71T^{2} \) |
| 73 | \( 1 - 15.5T + 73T^{2} \) |
| 79 | \( 1 + 2.17T + 79T^{2} \) |
| 83 | \( 1 - 2.48T + 83T^{2} \) |
| 89 | \( 1 + 1.95T + 89T^{2} \) |
| 97 | \( 1 - 15.4T + 97T^{2} \) |
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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
−8.547328665842078095766905394101, −7.66297346005149679432566842881, −7.36539881626559706859233045612, −6.33570599766962592161204869461, −5.56813113960552767593055668497, −4.12499251618407278954882696565, −3.78028809602676066942283726138, −3.04363404027642227704174061907, −2.30068956582852032439139812334, −0.852861542593058539916333028147,
0.852861542593058539916333028147, 2.30068956582852032439139812334, 3.04363404027642227704174061907, 3.78028809602676066942283726138, 4.12499251618407278954882696565, 5.56813113960552767593055668497, 6.33570599766962592161204869461, 7.36539881626559706859233045612, 7.66297346005149679432566842881, 8.547328665842078095766905394101