| L(s) = 1 | − 0.486·2-s − 0.821·3-s − 1.76·4-s + 5-s + 0.399·6-s + 3.94·7-s + 1.83·8-s − 2.32·9-s − 0.486·10-s + 4.13·11-s + 1.44·12-s − 0.310·13-s − 1.91·14-s − 0.821·15-s + 2.63·16-s + 1.37·17-s + 1.13·18-s − 1.76·20-s − 3.23·21-s − 2.01·22-s + 0.331·23-s − 1.50·24-s + 25-s + 0.151·26-s + 4.37·27-s − 6.95·28-s + 7.26·29-s + ⋯ |
| L(s) = 1 | − 0.344·2-s − 0.474·3-s − 0.881·4-s + 0.447·5-s + 0.163·6-s + 1.49·7-s + 0.647·8-s − 0.775·9-s − 0.153·10-s + 1.24·11-s + 0.417·12-s − 0.0862·13-s − 0.512·14-s − 0.212·15-s + 0.658·16-s + 0.332·17-s + 0.266·18-s − 0.394·20-s − 0.706·21-s − 0.429·22-s + 0.0691·23-s − 0.307·24-s + 0.200·25-s + 0.0296·26-s + 0.841·27-s − 1.31·28-s + 1.34·29-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
| \(L(1)\) |
\(\approx\) |
\(1.305964885\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.305964885\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.486T + 2T^{2} \) |
| 3 | \( 1 + 0.821T + 3T^{2} \) |
| 7 | \( 1 - 3.94T + 7T^{2} \) |
| 11 | \( 1 - 4.13T + 11T^{2} \) |
| 13 | \( 1 + 0.310T + 13T^{2} \) |
| 17 | \( 1 - 1.37T + 17T^{2} \) |
| 23 | \( 1 - 0.331T + 23T^{2} \) |
| 29 | \( 1 - 7.26T + 29T^{2} \) |
| 31 | \( 1 + 6.55T + 31T^{2} \) |
| 37 | \( 1 + 9.27T + 37T^{2} \) |
| 41 | \( 1 - 9.25T + 41T^{2} \) |
| 43 | \( 1 + 6.49T + 43T^{2} \) |
| 47 | \( 1 - 4.19T + 47T^{2} \) |
| 53 | \( 1 + 10.5T + 53T^{2} \) |
| 59 | \( 1 + 7.23T + 59T^{2} \) |
| 61 | \( 1 - 15.0T + 61T^{2} \) |
| 67 | \( 1 - 11.5T + 67T^{2} \) |
| 71 | \( 1 + 4.54T + 71T^{2} \) |
| 73 | \( 1 - 8.86T + 73T^{2} \) |
| 79 | \( 1 + 7.42T + 79T^{2} \) |
| 83 | \( 1 - 1.50T + 83T^{2} \) |
| 89 | \( 1 + 4.72T + 89T^{2} \) |
| 97 | \( 1 - 12.4T + 97T^{2} \) |
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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
−9.095412712508062878158727761986, −8.609254520881385280128722972632, −7.939452968857775107552672716871, −6.92280436711542803929694454182, −5.89522356963309634874996783163, −5.14773201337366913982483168139, −4.56199954702466646696495254437, −3.48639626682123805121078063304, −1.89201365623746652033578906039, −0.907115126695572334471266882801,
0.907115126695572334471266882801, 1.89201365623746652033578906039, 3.48639626682123805121078063304, 4.56199954702466646696495254437, 5.14773201337366913982483168139, 5.89522356963309634874996783163, 6.92280436711542803929694454182, 7.939452968857775107552672716871, 8.609254520881385280128722972632, 9.095412712508062878158727761986
