L(s) = 1 | − 0.719·2-s + 1.23·3-s − 1.48·4-s − 5-s − 0.888·6-s + 1.29·7-s + 2.50·8-s − 1.47·9-s + 0.719·10-s + 5.76·11-s − 1.82·12-s − 2.35·13-s − 0.929·14-s − 1.23·15-s + 1.16·16-s − 6.84·17-s + 1.06·18-s + 1.48·20-s + 1.59·21-s − 4.14·22-s + 5.66·23-s + 3.09·24-s + 25-s + 1.69·26-s − 5.52·27-s − 1.91·28-s + 6.02·29-s + ⋯ |
L(s) = 1 | − 0.508·2-s + 0.712·3-s − 0.741·4-s − 0.447·5-s − 0.362·6-s + 0.488·7-s + 0.885·8-s − 0.492·9-s + 0.227·10-s + 1.73·11-s − 0.528·12-s − 0.652·13-s − 0.248·14-s − 0.318·15-s + 0.290·16-s − 1.66·17-s + 0.250·18-s + 0.331·20-s + 0.347·21-s − 0.884·22-s + 1.18·23-s + 0.631·24-s + 0.200·25-s + 0.331·26-s − 1.06·27-s − 0.361·28-s + 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.316080853\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.316080853\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + T \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + 0.719T + 2T^{2} \) |
| 3 | \( 1 - 1.23T + 3T^{2} \) |
| 7 | \( 1 - 1.29T + 7T^{2} \) |
| 11 | \( 1 - 5.76T + 11T^{2} \) |
| 13 | \( 1 + 2.35T + 13T^{2} \) |
| 17 | \( 1 + 6.84T + 17T^{2} \) |
| 23 | \( 1 - 5.66T + 23T^{2} \) |
| 29 | \( 1 - 6.02T + 29T^{2} \) |
| 31 | \( 1 - 2.38T + 31T^{2} \) |
| 37 | \( 1 - 4.53T + 37T^{2} \) |
| 41 | \( 1 - 4.58T + 41T^{2} \) |
| 43 | \( 1 - 0.277T + 43T^{2} \) |
| 47 | \( 1 - 0.507T + 47T^{2} \) |
| 53 | \( 1 - 7.18T + 53T^{2} \) |
| 59 | \( 1 + 5.76T + 59T^{2} \) |
| 61 | \( 1 + 7.20T + 61T^{2} \) |
| 67 | \( 1 - 10.2T + 67T^{2} \) |
| 71 | \( 1 - 1.11T + 71T^{2} \) |
| 73 | \( 1 - 2.39T + 73T^{2} \) |
| 79 | \( 1 - 14.0T + 79T^{2} \) |
| 83 | \( 1 - 1.21T + 83T^{2} \) |
| 89 | \( 1 - 8.94T + 89T^{2} \) |
| 97 | \( 1 - 10.5T + 97T^{2} \) |
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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
−9.091871969313784235353911089947, −8.664195686645605017629367191000, −7.977413924033815840199597068914, −7.10617261002576670393133723167, −6.26553175558821796732965485570, −4.83631254614766587364580826432, −4.36293024442558735658502074257, −3.39420361806411458175355893421, −2.18075112209619640807878174253, −0.841075889633285071126227289892,
0.841075889633285071126227289892, 2.18075112209619640807878174253, 3.39420361806411458175355893421, 4.36293024442558735658502074257, 4.83631254614766587364580826432, 6.26553175558821796732965485570, 7.10617261002576670393133723167, 7.977413924033815840199597068914, 8.664195686645605017629367191000, 9.091871969313784235353911089947