L(s) = 1 | + 2-s − 2.33·3-s + 4-s − 5-s − 2.33·6-s + 4.65·7-s + 8-s + 2.45·9-s − 10-s − 4.33·11-s − 2.33·12-s − 6.61·13-s + 4.65·14-s + 2.33·15-s + 16-s + 3.73·17-s + 2.45·18-s − 6.87·19-s − 20-s − 10.8·21-s − 4.33·22-s − 4.47·23-s − 2.33·24-s + 25-s − 6.61·26-s + 1.27·27-s + 4.65·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.34·3-s + 0.5·4-s − 0.447·5-s − 0.953·6-s + 1.75·7-s + 0.353·8-s + 0.818·9-s − 0.316·10-s − 1.30·11-s − 0.674·12-s − 1.83·13-s + 1.24·14-s + 0.603·15-s + 0.250·16-s + 0.905·17-s + 0.578·18-s − 1.57·19-s − 0.223·20-s − 2.37·21-s − 0.924·22-s − 0.932·23-s − 0.476·24-s + 0.200·25-s − 1.29·26-s + 0.244·27-s + 0.879·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.411457425\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.411457425\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 601 | \( 1 + T \) |
good | 3 | \( 1 + 2.33T + 3T^{2} \) |
| 7 | \( 1 - 4.65T + 7T^{2} \) |
| 11 | \( 1 + 4.33T + 11T^{2} \) |
| 13 | \( 1 + 6.61T + 13T^{2} \) |
| 17 | \( 1 - 3.73T + 17T^{2} \) |
| 19 | \( 1 + 6.87T + 19T^{2} \) |
| 23 | \( 1 + 4.47T + 23T^{2} \) |
| 29 | \( 1 - 0.414T + 29T^{2} \) |
| 31 | \( 1 - 1.39T + 31T^{2} \) |
| 37 | \( 1 - 3.22T + 37T^{2} \) |
| 41 | \( 1 - 12.5T + 41T^{2} \) |
| 43 | \( 1 + 6.42T + 43T^{2} \) |
| 47 | \( 1 + 4.28T + 47T^{2} \) |
| 53 | \( 1 + 6.31T + 53T^{2} \) |
| 59 | \( 1 - 13.8T + 59T^{2} \) |
| 61 | \( 1 - 3.24T + 61T^{2} \) |
| 67 | \( 1 - 5.70T + 67T^{2} \) |
| 71 | \( 1 - 3.34T + 71T^{2} \) |
| 73 | \( 1 - 10.6T + 73T^{2} \) |
| 79 | \( 1 + 1.61T + 79T^{2} \) |
| 83 | \( 1 - 13.6T + 83T^{2} \) |
| 89 | \( 1 + 18.2T + 89T^{2} \) |
| 97 | \( 1 + 2.63T + 97T^{2} \) |
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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
−7.957625457553571629661871904358, −7.36654104974488922632225601751, −6.52333537723588494406484198888, −5.61523319326127330423339112064, −5.15236970056426934051110757592, −4.70020006894763968759236193617, −4.10696127596796656266580592459, −2.62765251691312836198181917332, −1.98740209541844253549771552750, −0.58028303919615581742872289126,
0.58028303919615581742872289126, 1.98740209541844253549771552750, 2.62765251691312836198181917332, 4.10696127596796656266580592459, 4.70020006894763968759236193617, 5.15236970056426934051110757592, 5.61523319326127330423339112064, 6.52333537723588494406484198888, 7.36654104974488922632225601751, 7.957625457553571629661871904358