| L(s) = 1 | + 1.36·2-s + 3-s − 0.149·4-s − 2.92·5-s + 1.36·6-s − 2.92·8-s + 9-s − 3.97·10-s − 11-s − 0.149·12-s + 6.54·13-s − 2.92·15-s − 3.67·16-s + 8.05·17-s + 1.36·18-s + 0.279·19-s + 0.436·20-s − 1.36·22-s + 1.78·23-s − 2.92·24-s + 3.54·25-s + 8.90·26-s + 27-s + 3.90·29-s − 3.97·30-s − 0.617·31-s + 0.842·32-s + ⋯ |
| L(s) = 1 | + 0.961·2-s + 0.577·3-s − 0.0746·4-s − 1.30·5-s + 0.555·6-s − 1.03·8-s + 0.333·9-s − 1.25·10-s − 0.301·11-s − 0.0430·12-s + 1.81·13-s − 0.754·15-s − 0.919·16-s + 1.95·17-s + 0.320·18-s + 0.0640·19-s + 0.0976·20-s − 0.290·22-s + 0.373·23-s − 0.596·24-s + 0.709·25-s + 1.74·26-s + 0.192·27-s + 0.725·29-s − 0.726·30-s − 0.110·31-s + 0.148·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.521158198\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.521158198\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 - 1.36T + 2T^{2} \) |
| 5 | \( 1 + 2.92T + 5T^{2} \) |
| 13 | \( 1 - 6.54T + 13T^{2} \) |
| 17 | \( 1 - 8.05T + 17T^{2} \) |
| 19 | \( 1 - 0.279T + 19T^{2} \) |
| 23 | \( 1 - 1.78T + 23T^{2} \) |
| 29 | \( 1 - 3.90T + 29T^{2} \) |
| 31 | \( 1 + 0.617T + 31T^{2} \) |
| 37 | \( 1 + 4.25T + 37T^{2} \) |
| 41 | \( 1 - 2.88T + 41T^{2} \) |
| 43 | \( 1 + 4.64T + 43T^{2} \) |
| 47 | \( 1 - 12.4T + 47T^{2} \) |
| 53 | \( 1 - 7.44T + 53T^{2} \) |
| 59 | \( 1 - 10.7T + 59T^{2} \) |
| 61 | \( 1 + 15.3T + 61T^{2} \) |
| 67 | \( 1 - 4.69T + 67T^{2} \) |
| 71 | \( 1 - 9.38T + 71T^{2} \) |
| 73 | \( 1 + 0.847T + 73T^{2} \) |
| 79 | \( 1 + 5.23T + 79T^{2} \) |
| 83 | \( 1 - 7.36T + 83T^{2} \) |
| 89 | \( 1 + 6.99T + 89T^{2} \) |
| 97 | \( 1 + 8.07T + 97T^{2} \) |
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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
−9.182004213707835204257853094629, −8.438735089631204700813709701769, −7.953063870095829657372254268628, −7.01655890927004657086157591111, −5.93503697147034666003793563194, −5.17366812002356849088380308740, −4.03164980247202405204128974197, −3.64977155311018948776278480653, −2.91540576652191457277819809456, −0.986522990278235136688963556835,
0.986522990278235136688963556835, 2.91540576652191457277819809456, 3.64977155311018948776278480653, 4.03164980247202405204128974197, 5.17366812002356849088380308740, 5.93503697147034666003793563194, 7.01655890927004657086157591111, 7.953063870095829657372254268628, 8.438735089631204700813709701769, 9.182004213707835204257853094629
