| L(s) = 1 | − 2·2-s − 3·3-s + 4·4-s + 5·5-s + 6·6-s − 8·8-s + 9·9-s − 10·10-s − 6.52·11-s − 12·12-s + 16.9·13-s − 15·15-s + 16·16-s + 117.·17-s − 18·18-s + 113.·19-s + 20·20-s + 13.0·22-s − 177.·23-s + 24·24-s + 25·25-s − 33.8·26-s − 27·27-s + 230.·29-s + 30·30-s − 306.·31-s − 32·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s − 0.353·8-s + 0.333·9-s − 0.316·10-s − 0.178·11-s − 0.288·12-s + 0.361·13-s − 0.258·15-s + 0.250·16-s + 1.67·17-s − 0.235·18-s + 1.36·19-s + 0.223·20-s + 0.126·22-s − 1.60·23-s + 0.204·24-s + 0.200·25-s − 0.255·26-s − 0.192·27-s + 1.47·29-s + 0.182·30-s − 1.77·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.455490943\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.455490943\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 + 3T \) |
| 5 | \( 1 - 5T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 6.52T + 1.33e3T^{2} \) |
| 13 | \( 1 - 16.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 117.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 113.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 177.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 230.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 306.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 355.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 119.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 395.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 411.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 697.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 158.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 256.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 675.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 457.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 651.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 152T + 4.93e5T^{2} \) |
| 83 | \( 1 + 811.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 657.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.42e3T + 9.12e5T^{2} \) |
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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
−9.336966699081090660758272621256, −8.250193898491134162397311979532, −7.62583047494636630813710964624, −6.76844867471864814739238679126, −5.75382183712185734036917071533, −5.39375232189804899949338198421, −3.96068395772349139824160139283, −2.89163286686456329481244948126, −1.60824305236788146370921990300, −0.71912833505289888278488148277,
0.71912833505289888278488148277, 1.60824305236788146370921990300, 2.89163286686456329481244948126, 3.96068395772349139824160139283, 5.39375232189804899949338198421, 5.75382183712185734036917071533, 6.76844867471864814739238679126, 7.62583047494636630813710964624, 8.250193898491134162397311979532, 9.336966699081090660758272621256
