L(s) = 1 | + 0.414·2-s − 1.41·3-s − 1.82·4-s − 5-s − 0.585·6-s + 7-s − 1.58·8-s − 0.999·9-s − 0.414·10-s + 2.58·12-s − 3.41·13-s + 0.414·14-s + 1.41·15-s + 3·16-s + 0.585·17-s − 0.414·18-s + 1.82·20-s − 1.41·21-s + 8.82·23-s + 2.24·24-s + 25-s − 1.41·26-s + 5.65·27-s − 1.82·28-s + 0.828·29-s + 0.585·30-s − 1.75·31-s + ⋯ |
L(s) = 1 | + 0.292·2-s − 0.816·3-s − 0.914·4-s − 0.447·5-s − 0.239·6-s + 0.377·7-s − 0.560·8-s − 0.333·9-s − 0.130·10-s + 0.746·12-s − 0.946·13-s + 0.110·14-s + 0.365·15-s + 0.750·16-s + 0.142·17-s − 0.0976·18-s + 0.408·20-s − 0.308·21-s + 1.84·23-s + 0.457·24-s + 0.200·25-s − 0.277·26-s + 1.08·27-s − 0.345·28-s + 0.153·29-s + 0.106·30-s − 0.315·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - 0.414T + 2T^{2} \) |
| 3 | \( 1 + 1.41T + 3T^{2} \) |
| 13 | \( 1 + 3.41T + 13T^{2} \) |
| 17 | \( 1 - 0.585T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 8.82T + 23T^{2} \) |
| 29 | \( 1 - 0.828T + 29T^{2} \) |
| 31 | \( 1 + 1.75T + 31T^{2} \) |
| 37 | \( 1 - 1.17T + 37T^{2} \) |
| 41 | \( 1 + 4.24T + 41T^{2} \) |
| 43 | \( 1 + 6T + 43T^{2} \) |
| 47 | \( 1 - 1.41T + 47T^{2} \) |
| 53 | \( 1 - 2.82T + 53T^{2} \) |
| 59 | \( 1 + 7.89T + 59T^{2} \) |
| 61 | \( 1 - 13.8T + 61T^{2} \) |
| 67 | \( 1 - 6.48T + 67T^{2} \) |
| 71 | \( 1 - 9.17T + 71T^{2} \) |
| 73 | \( 1 - 5.07T + 73T^{2} \) |
| 79 | \( 1 + 6.48T + 79T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 - 0.828T + 89T^{2} \) |
| 97 | \( 1 - 9.31T + 97T^{2} \) |
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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
−8.149132046738448833512900903605, −7.19928930490002872794976840561, −6.54432421715936857997355440908, −5.41722959807774290876095240514, −5.15970965587574463859018826269, −4.48086834374553856740265055790, −3.51603340988850335931523430449, −2.65373126249656688844426325396, −1.03955641092666788775615291438, 0,
1.03955641092666788775615291438, 2.65373126249656688844426325396, 3.51603340988850335931523430449, 4.48086834374553856740265055790, 5.15970965587574463859018826269, 5.41722959807774290876095240514, 6.54432421715936857997355440908, 7.19928930490002872794976840561, 8.149132046738448833512900903605